Solve x^2+ax+b/x^2+7x+12*x^2+3x-4/x^2-1=x+2/x+3quada-b=?

$$\frac { x ^ { 2 } + a x + b } { x ^ { 2 } + 7 x + 12 } \cdot \frac { x ^ { 2 } + 3 x - 4 } { x ^ { 2 } - 1 } = \frac { x + 2 } { x + 3 } \quad a - b = ?$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{bx^{2}+x^{2}+4bx+4b}{x^{2}+2x+2}\text{, }&x\neq -1-i\text{ and }x\neq -1+i\text{ and }x\neq -1\text{ and }x\neq -3\text{ and }x\neq -4\text{ and }x\neq 1\\a\in \mathrm{C}\text{, }&\left(x=-1+i\text{ or }x=-1-i\right)\text{ and }b=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right), the least common multiple of x^{2}+7x+12,x^{2}-1,x+3.

$$\left(x^{2}-1\right)\left(x^{2}+ax+b\right)\times \frac{x^{2}+3x-4}{x^{2}-1}=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{x^{2}+3x-4}{x^{2}-1}.

$$\left(x^{2}-1\right)\left(x^{2}+ax+b\right)\times \frac{\left(x-1\right)\left(x+4\right)}{\left(x-1\right)\left(x+1\right)}=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x-1 in both numerator and denominator.

$$\left(x^{2}-1\right)\left(x^{2}+ax+b\right)\times \frac{x+4}{x+1}=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Express \left(x^{2}-1\right)\times \frac{x+4}{x+1} as a single fraction.

$$\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}\left(x^{2}+ax+b\right)=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1} by x^{2}+ax+b.

$$\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}.

$$\frac{\left(x-1\right)\left(x+1\right)\left(x+4\right)}{x+1}x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x+1 in both numerator and denominator.

$$\left(x-1\right)\left(x+4\right)x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Expand the expression.

$$\left(x^{2}+3x-4\right)x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}+3x-4 by x^{2}.

$$x^{4}+3x^{3}-4x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}.

$$x^{4}+3x^{3}-4x^{2}+\frac{\left(x-1\right)\left(x+1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x+1 in both numerator and denominator.

$$x^{4}+3x^{3}-4x^{2}+\left(x-1\right)\left(x+4\right)ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Expand the expression.

$$x^{4}+3x^{3}-4x^{2}+\left(x^{2}+3x-4\right)ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}+3x-4 by a.

$$x^{4}+3x^{3}-4x^{2}+\left(x^{2}a+3xa-4a\right)x+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}a+3xa-4a by x.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\frac{\left(x-1\right)\left(x+1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x+1 in both numerator and denominator.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\left(x-1\right)\left(x+4\right)b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Expand the expression.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\left(x^{2}+3x-4\right)b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}+3x-4 by b.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x+4 by x^{2}-1.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=\left(x^{3}-x+4x^{2}-4\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{3}-x+4x^{2}-4 by x+2 and combine like terms.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=\left(x^{4}+6x^{3}+7x^{2}-6x-8\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{4}+6x^{3}+7x^{2}-6x-8 by a.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply -b by x-1.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+\left(-bx+b\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply -bx+b by x+1 and combine like terms.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+\left(-bx^{2}+b\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply -bx^{2}+b by x+3.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+\left(-bx^{3}-3bx^{2}+bx+3b\right)\left(x+4\right)$$

Use the distributive property to multiply -bx^{3}-3bx^{2}+bx+3b by x+4 and combine like terms.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Subtract x^{4}a from both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b-x^{4}a=6x^{3}a+7x^{2}a-6xa-8a-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Subtract 6x^{3}a from both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b-x^{4}a-6x^{3}a=7x^{2}a-6xa-8a-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Subtract 7x^{2}a from both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b-x^{4}a-6x^{3}a-7x^{2}a=-6xa-8a-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Add 6xa to both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b-x^{4}a-6x^{3}a-7x^{2}a+6xa=-8a-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Add 8a to both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b-x^{4}a-6x^{3}a-7x^{2}a+6xa+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Combine ax^{3} and -6x^{3}a to get -5ax^{3}.

$$x^{4}+3x^{3}-4x^{2}-5ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b-x^{4}a-7x^{2}a+6xa+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Combine 3ax^{2} and -7x^{2}a to get -4ax^{2}.

$$x^{4}+3x^{3}-4x^{2}-5ax^{3}-4ax^{2}-4ax+x^{2}b+3xb-4b-x^{4}a+6xa+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Combine -4ax and 6xa to get 2ax.

$$x^{4}+3x^{3}-4x^{2}-5ax^{3}-4ax^{2}+2ax+x^{2}b+3xb-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Subtract x^{4} from both sides.

$$3x^{3}-4x^{2}-5ax^{3}-4ax^{2}+2ax+x^{2}b+3xb-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b-x^{4}$$

Subtract 3x^{3} from both sides.

$$-4x^{2}-5ax^{3}-4ax^{2}+2ax+x^{2}b+3xb-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b-x^{4}-3x^{3}$$

Add 4x^{2} to both sides.

$$-5ax^{3}-4ax^{2}+2ax+x^{2}b+3xb-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b-x^{4}-3x^{3}+4x^{2}$$

Subtract x^{2}b from both sides.

$$-5ax^{3}-4ax^{2}+2ax+3xb-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b-x^{4}-3x^{3}+4x^{2}-x^{2}b$$

Combine -11bx^{2} and -x^{2}b to get -12bx^{2}.

$$-5ax^{3}-4ax^{2}+2ax+3xb-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-12bx^{2}+7bx+12b-x^{4}-3x^{3}+4x^{2}$$

Subtract 3xb from both sides.

$$-5ax^{3}-4ax^{2}+2ax-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-12bx^{2}+7bx+12b-x^{4}-3x^{3}+4x^{2}-3xb$$

Combine 7bx and -3xb to get 4bx.

$$-5ax^{3}-4ax^{2}+2ax-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-12bx^{2}+4bx+12b-x^{4}-3x^{3}+4x^{2}$$

Add 4b to both sides.

$$-5ax^{3}-4ax^{2}+2ax-x^{4}a+8a=-bx^{4}-7bx^{3}-12bx^{2}+4bx+12b-x^{4}-3x^{3}+4x^{2}+4b$$

Combine 12b and 4b to get 16b.

$$-5ax^{3}-4ax^{2}+2ax-x^{4}a+8a=-bx^{4}-7bx^{3}-12bx^{2}+4bx+16b-x^{4}-3x^{3}+4x^{2}$$

Combine all terms containing a.

$$\left(-5x^{3}-4x^{2}+2x-x^{4}+8\right)a=-bx^{4}-7bx^{3}-12bx^{2}+4bx+16b-x^{4}-3x^{3}+4x^{2}$$

The equation is in standard form.

$$\left(8+2x-4x^{2}-5x^{3}-x^{4}\right)a=16b+4bx+4x^{2}-12bx^{2}-3x^{3}-7bx^{3}-x^{4}-bx^{4}$$

Divide both sides by -5x^{3}-4x^{2}+2x-x^{4}+8.

$$\frac{\left(8+2x-4x^{2}-5x^{3}-x^{4}\right)a}{8+2x-4x^{2}-5x^{3}-x^{4}}=-\frac{\left(x-1\right)\left(x+4\right)\left(bx^{2}+x^{2}+4bx+4b\right)}{8+2x-4x^{2}-5x^{3}-x^{4}}$$

Dividing by -5x^{3}-4x^{2}+2x-x^{4}+8 undoes the multiplication by -5x^{3}-4x^{2}+2x-x^{4}+8.

$$a=-\frac{\left(x-1\right)\left(x+4\right)\left(bx^{2}+x^{2}+4bx+4b\right)}{8+2x-4x^{2}-5x^{3}-x^{4}}$$

Divide -\left(-1+x\right)\left(4+x\right)\left(x^{2}+4b+4bx+bx^{2}\right) by -5x^{3}-4x^{2}+2x-x^{4}+8.

$$a=\frac{bx^{2}+x^{2}+4bx+4b}{x^{2}+2x+2}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{-ax^{2}+x^{2}-2ax-2a}{\left(x+2\right)^{2}}\text{, }&x\neq -2\text{ and }x\neq -1\text{ and }x\neq -3\text{ and }x\neq -4\text{ and }x\neq 1\\b\in \mathrm{C}\text{, }&x=-2\text{ and }a=2\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right), the least common multiple of x^{2}+7x+12,x^{2}-1,x+3.

$$\left(x^{2}-1\right)\left(x^{2}+ax+b\right)\times \frac{x^{2}+3x-4}{x^{2}-1}=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{x^{2}+3x-4}{x^{2}-1}.

$$\left(x^{2}-1\right)\left(x^{2}+ax+b\right)\times \frac{\left(x-1\right)\left(x+4\right)}{\left(x-1\right)\left(x+1\right)}=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x-1 in both numerator and denominator.

$$\left(x^{2}-1\right)\left(x^{2}+ax+b\right)\times \frac{x+4}{x+1}=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Express \left(x^{2}-1\right)\times \frac{x+4}{x+1} as a single fraction.

$$\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}\left(x^{2}+ax+b\right)=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1} by x^{2}+ax+b.

$$\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}.

$$\frac{\left(x-1\right)\left(x+1\right)\left(x+4\right)}{x+1}x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x+1 in both numerator and denominator.

$$\left(x-1\right)\left(x+4\right)x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Expand the expression.

$$\left(x^{2}+3x-4\right)x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}+3x-4 by x^{2}.

$$x^{4}+3x^{3}-4x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}.

$$x^{4}+3x^{3}-4x^{2}+\frac{\left(x-1\right)\left(x+1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x+1 in both numerator and denominator.

$$x^{4}+3x^{3}-4x^{2}+\left(x-1\right)\left(x+4\right)ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Expand the expression.

$$x^{4}+3x^{3}-4x^{2}+\left(x^{2}+3x-4\right)ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}+3x-4 by a.

$$x^{4}+3x^{3}-4x^{2}+\left(x^{2}a+3xa-4a\right)x+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}a+3xa-4a by x.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\frac{\left(x-1\right)\left(x+1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x+1 in both numerator and denominator.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\left(x-1\right)\left(x+4\right)b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Expand the expression.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\left(x^{2}+3x-4\right)b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}+3x-4 by b.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x+4 by x^{2}-1.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=\left(x^{3}-x+4x^{2}-4\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{3}-x+4x^{2}-4 by x+2 and combine like terms.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=\left(x^{4}+6x^{3}+7x^{2}-6x-8\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{4}+6x^{3}+7x^{2}-6x-8 by a.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply -b by x-1.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+\left(-bx+b\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply -bx+b by x+1 and combine like terms.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+\left(-bx^{2}+b\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply -bx^{2}+b by x+3.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+\left(-bx^{3}-3bx^{2}+bx+3b\right)\left(x+4\right)$$

Use the distributive property to multiply -bx^{3}-3bx^{2}+bx+3b by x+4 and combine like terms.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Add bx^{4} to both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b+bx^{4}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-7bx^{3}-11bx^{2}+7bx+12b$$

Add 7bx^{3} to both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-11bx^{2}+7bx+12b$$

Add 11bx^{2} to both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b+bx^{4}+7bx^{3}+11bx^{2}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+7bx+12b$$

Combine x^{2}b and 11bx^{2} to get 12x^{2}b.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+12x^{2}b+3xb-4b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+7bx+12b$$

Subtract 7bx from both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+12x^{2}b+3xb-4b+bx^{4}+7bx^{3}-7bx=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+12b$$

Combine 3xb and -7bx to get -4xb.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+12x^{2}b-4xb-4b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+12b$$

Subtract 12b from both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+12x^{2}b-4xb-4b+bx^{4}+7bx^{3}-12b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a$$

Combine -4b and -12b to get -16b.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a$$

Subtract x^{4} from both sides.

$$3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-x^{4}$$

Subtract 3x^{3} from both sides.

$$-4x^{2}+ax^{3}+3ax^{2}-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-x^{4}-3x^{3}$$

Add 4x^{2} to both sides.

$$ax^{3}+3ax^{2}-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-x^{4}-3x^{3}+4x^{2}$$

Subtract ax^{3} from both sides.

$$3ax^{2}-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-x^{4}-3x^{3}+4x^{2}-ax^{3}$$

Combine 6x^{3}a and -ax^{3} to get 5x^{3}a.

$$3ax^{2}-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+5x^{3}a+7x^{2}a-6xa-8a-x^{4}-3x^{3}+4x^{2}$$

Subtract 3ax^{2} from both sides.

$$-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+5x^{3}a+7x^{2}a-6xa-8a-x^{4}-3x^{3}+4x^{2}-3ax^{2}$$

Combine 7x^{2}a and -3ax^{2} to get 4x^{2}a.

$$-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+5x^{3}a+4x^{2}a-6xa-8a-x^{4}-3x^{3}+4x^{2}$$

Add 4ax to both sides.

$$12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+5x^{3}a+4x^{2}a-6xa-8a-x^{4}-3x^{3}+4x^{2}+4ax$$

Combine -6xa and 4ax to get -2xa.

$$12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+5x^{3}a+4x^{2}a-2xa-8a-x^{4}-3x^{3}+4x^{2}$$

Combine all terms containing b.

$$\left(12x^{2}-4x-16+x^{4}+7x^{3}\right)b=x^{4}a+5x^{3}a+4x^{2}a-2xa-8a-x^{4}-3x^{3}+4x^{2}$$

The equation is in standard form.

$$\left(x^{4}+7x^{3}+12x^{2}-4x-16\right)b=ax^{4}-x^{4}+5ax^{3}-3x^{3}+4ax^{2}+4x^{2}-2ax-8a$$

Divide both sides by 12x^{2}-4x-16+x^{4}+7x^{3}.

$$\frac{\left(x^{4}+7x^{3}+12x^{2}-4x-16\right)b}{x^{4}+7x^{3}+12x^{2}-4x-16}=\frac{\left(x-1\right)\left(x+4\right)\left(ax^{2}-x^{2}+2ax+2a\right)}{x^{4}+7x^{3}+12x^{2}-4x-16}$$

Dividing by 12x^{2}-4x-16+x^{4}+7x^{3} undoes the multiplication by 12x^{2}-4x-16+x^{4}+7x^{3}.

$$b=\frac{\left(x-1\right)\left(x+4\right)\left(ax^{2}-x^{2}+2ax+2a\right)}{x^{4}+7x^{3}+12x^{2}-4x-16}$$

Divide \left(-1+x\right)\left(4+x\right)\left(-x^{2}+2a+2xa+x^{2}a\right) by 12x^{2}-4x-16+x^{4}+7x^{3}.

$$b=\frac{ax^{2}-x^{2}+2ax+2a}{\left(x+2\right)^{2}}$$

Solve for a

$$a=\frac{bx^{2}+x^{2}+4bx+4b}{x^{2}+2x+2}$$

$$x\neq -3\text{ and }x\neq -4\text{ and }|x|\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right), the least common multiple of x^{2}+7x+12,x^{2}-1,x+3.

$$\left(x^{2}-1\right)\left(x^{2}+ax+b\right)\times \frac{x^{2}+3x-4}{x^{2}-1}=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{x^{2}+3x-4}{x^{2}-1}.

$$\left(x^{2}-1\right)\left(x^{2}+ax+b\right)\times \frac{\left(x-1\right)\left(x+4\right)}{\left(x-1\right)\left(x+1\right)}=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x-1 in both numerator and denominator.

$$\left(x^{2}-1\right)\left(x^{2}+ax+b\right)\times \frac{x+4}{x+1}=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Express \left(x^{2}-1\right)\times \frac{x+4}{x+1} as a single fraction.

$$\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}\left(x^{2}+ax+b\right)=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1} by x^{2}+ax+b.

$$\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}.

$$\frac{\left(x-1\right)\left(x+1\right)\left(x+4\right)}{x+1}x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x+1 in both numerator and denominator.

$$\left(x-1\right)\left(x+4\right)x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Expand the expression.

$$\left(x^{2}+3x-4\right)x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}+3x-4 by x^{2}.

$$x^{4}+3x^{3}-4x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}.

$$x^{4}+3x^{3}-4x^{2}+\frac{\left(x-1\right)\left(x+1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x+1 in both numerator and denominator.

$$x^{4}+3x^{3}-4x^{2}+\left(x-1\right)\left(x+4\right)ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Expand the expression.

$$x^{4}+3x^{3}-4x^{2}+\left(x^{2}+3x-4\right)ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}+3x-4 by a.

$$x^{4}+3x^{3}-4x^{2}+\left(x^{2}a+3xa-4a\right)x+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}a+3xa-4a by x.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\frac{\left(x-1\right)\left(x+1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x+1 in both numerator and denominator.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\left(x-1\right)\left(x+4\right)b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Expand the expression.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\left(x^{2}+3x-4\right)b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}+3x-4 by b.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x+4 by x^{2}-1.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=\left(x^{3}-x+4x^{2}-4\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{3}-x+4x^{2}-4 by x+2 and combine like terms.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=\left(x^{4}+6x^{3}+7x^{2}-6x-8\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{4}+6x^{3}+7x^{2}-6x-8 by a.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply -b by x-1.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+\left(-bx+b\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply -bx+b by x+1 and combine like terms.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+\left(-bx^{2}+b\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply -bx^{2}+b by x+3.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+\left(-bx^{3}-3bx^{2}+bx+3b\right)\left(x+4\right)$$

Use the distributive property to multiply -bx^{3}-3bx^{2}+bx+3b by x+4 and combine like terms.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Subtract x^{4}a from both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b-x^{4}a=6x^{3}a+7x^{2}a-6xa-8a-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Subtract 6x^{3}a from both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b-x^{4}a-6x^{3}a=7x^{2}a-6xa-8a-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Subtract 7x^{2}a from both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b-x^{4}a-6x^{3}a-7x^{2}a=-6xa-8a-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Add 6xa to both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b-x^{4}a-6x^{3}a-7x^{2}a+6xa=-8a-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Add 8a to both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b-x^{4}a-6x^{3}a-7x^{2}a+6xa+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Combine ax^{3} and -6x^{3}a to get -5ax^{3}.

$$x^{4}+3x^{3}-4x^{2}-5ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b-x^{4}a-7x^{2}a+6xa+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Combine 3ax^{2} and -7x^{2}a to get -4ax^{2}.

$$x^{4}+3x^{3}-4x^{2}-5ax^{3}-4ax^{2}-4ax+x^{2}b+3xb-4b-x^{4}a+6xa+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Combine -4ax and 6xa to get 2ax.

$$x^{4}+3x^{3}-4x^{2}-5ax^{3}-4ax^{2}+2ax+x^{2}b+3xb-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Subtract x^{4} from both sides.

$$3x^{3}-4x^{2}-5ax^{3}-4ax^{2}+2ax+x^{2}b+3xb-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b-x^{4}$$

Subtract 3x^{3} from both sides.

$$-4x^{2}-5ax^{3}-4ax^{2}+2ax+x^{2}b+3xb-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b-x^{4}-3x^{3}$$

Add 4x^{2} to both sides.

$$-5ax^{3}-4ax^{2}+2ax+x^{2}b+3xb-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b-x^{4}-3x^{3}+4x^{2}$$

Subtract x^{2}b from both sides.

$$-5ax^{3}-4ax^{2}+2ax+3xb-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b-x^{4}-3x^{3}+4x^{2}-x^{2}b$$

Combine -11bx^{2} and -x^{2}b to get -12bx^{2}.

$$-5ax^{3}-4ax^{2}+2ax+3xb-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-12bx^{2}+7bx+12b-x^{4}-3x^{3}+4x^{2}$$

Subtract 3xb from both sides.

$$-5ax^{3}-4ax^{2}+2ax-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-12bx^{2}+7bx+12b-x^{4}-3x^{3}+4x^{2}-3xb$$

Combine 7bx and -3xb to get 4bx.

$$-5ax^{3}-4ax^{2}+2ax-4b-x^{4}a+8a=-bx^{4}-7bx^{3}-12bx^{2}+4bx+12b-x^{4}-3x^{3}+4x^{2}$$

Add 4b to both sides.

$$-5ax^{3}-4ax^{2}+2ax-x^{4}a+8a=-bx^{4}-7bx^{3}-12bx^{2}+4bx+12b-x^{4}-3x^{3}+4x^{2}+4b$$

Combine 12b and 4b to get 16b.

$$-5ax^{3}-4ax^{2}+2ax-x^{4}a+8a=-bx^{4}-7bx^{3}-12bx^{2}+4bx+16b-x^{4}-3x^{3}+4x^{2}$$

Combine all terms containing a.

$$\left(-5x^{3}-4x^{2}+2x-x^{4}+8\right)a=-bx^{4}-7bx^{3}-12bx^{2}+4bx+16b-x^{4}-3x^{3}+4x^{2}$$

The equation is in standard form.

$$\left(8+2x-4x^{2}-5x^{3}-x^{4}\right)a=16b+4bx+4x^{2}-12bx^{2}-3x^{3}-7bx^{3}-x^{4}-bx^{4}$$

Divide both sides by -5x^{3}-4x^{2}+2x-x^{4}+8.

$$\frac{\left(8+2x-4x^{2}-5x^{3}-x^{4}\right)a}{8+2x-4x^{2}-5x^{3}-x^{4}}=-\frac{\left(x-1\right)\left(x+4\right)\left(bx^{2}+x^{2}+4bx+4b\right)}{8+2x-4x^{2}-5x^{3}-x^{4}}$$

Dividing by -5x^{3}-4x^{2}+2x-x^{4}+8 undoes the multiplication by -5x^{3}-4x^{2}+2x-x^{4}+8.

$$a=-\frac{\left(x-1\right)\left(x+4\right)\left(bx^{2}+x^{2}+4bx+4b\right)}{8+2x-4x^{2}-5x^{3}-x^{4}}$$

Divide -\left(-1+x\right)\left(4+x\right)\left(x^{2}+4b+4bx+bx^{2}\right) by -5x^{3}-4x^{2}+2x-x^{4}+8.

$$a=\frac{bx^{2}+x^{2}+4bx+4b}{x^{2}+2x+2}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{-ax^{2}+x^{2}-2ax-2a}{\left(x+2\right)^{2}}\text{, }&x\neq -2\text{ and }x\neq -3\text{ and }x\neq -4\text{ and }|x|\neq 1\\b\in \mathrm{R}\text{, }&x=-2\text{ and }a=2\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right), the least common multiple of x^{2}+7x+12,x^{2}-1,x+3.

$$\left(x^{2}-1\right)\left(x^{2}+ax+b\right)\times \frac{x^{2}+3x-4}{x^{2}-1}=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{x^{2}+3x-4}{x^{2}-1}.

$$\left(x^{2}-1\right)\left(x^{2}+ax+b\right)\times \frac{\left(x-1\right)\left(x+4\right)}{\left(x-1\right)\left(x+1\right)}=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x-1 in both numerator and denominator.

$$\left(x^{2}-1\right)\left(x^{2}+ax+b\right)\times \frac{x+4}{x+1}=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Express \left(x^{2}-1\right)\times \frac{x+4}{x+1} as a single fraction.

$$\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}\left(x^{2}+ax+b\right)=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1} by x^{2}+ax+b.

$$\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}.

$$\frac{\left(x-1\right)\left(x+1\right)\left(x+4\right)}{x+1}x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x+1 in both numerator and denominator.

$$\left(x-1\right)\left(x+4\right)x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Expand the expression.

$$\left(x^{2}+3x-4\right)x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}+3x-4 by x^{2}.

$$x^{4}+3x^{3}-4x^{2}+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}.

$$x^{4}+3x^{3}-4x^{2}+\frac{\left(x-1\right)\left(x+1\right)\left(x+4\right)}{x+1}ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x+1 in both numerator and denominator.

$$x^{4}+3x^{3}-4x^{2}+\left(x-1\right)\left(x+4\right)ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Expand the expression.

$$x^{4}+3x^{3}-4x^{2}+\left(x^{2}+3x-4\right)ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}+3x-4 by a.

$$x^{4}+3x^{3}-4x^{2}+\left(x^{2}a+3xa-4a\right)x+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}a+3xa-4a by x.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Factor the expressions that are not already factored in \frac{\left(x^{2}-1\right)\left(x+4\right)}{x+1}.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\frac{\left(x-1\right)\left(x+1\right)\left(x+4\right)}{x+1}b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Cancel out x+1 in both numerator and denominator.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\left(x-1\right)\left(x+4\right)b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Expand the expression.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+\left(x^{2}+3x-4\right)b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}+3x-4 by b.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=\left(x+4\right)\left(x^{2}-1\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x+4 by x^{2}-1.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=\left(x^{3}-x+4x^{2}-4\right)\left(x+2\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{3}-x+4x^{2}-4 by x+2 and combine like terms.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=\left(x^{4}+6x^{3}+7x^{2}-6x-8\right)a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply x^{4}+6x^{3}+7x^{2}-6x-8 by a.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-b\left(x-1\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply -b by x-1.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+\left(-bx+b\right)\left(x+1\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply -bx+b by x+1 and combine like terms.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+\left(-bx^{2}+b\right)\left(x+3\right)\left(x+4\right)$$

Use the distributive property to multiply -bx^{2}+b by x+3.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+\left(-bx^{3}-3bx^{2}+bx+3b\right)\left(x+4\right)$$

Use the distributive property to multiply -bx^{3}-3bx^{2}+bx+3b by x+4 and combine like terms.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-bx^{4}-7bx^{3}-11bx^{2}+7bx+12b$$

Add bx^{4} to both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b+bx^{4}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-7bx^{3}-11bx^{2}+7bx+12b$$

Add 7bx^{3} to both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-11bx^{2}+7bx+12b$$

Add 11bx^{2} to both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+x^{2}b+3xb-4b+bx^{4}+7bx^{3}+11bx^{2}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+7bx+12b$$

Combine x^{2}b and 11bx^{2} to get 12x^{2}b.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+12x^{2}b+3xb-4b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+7bx+12b$$

Subtract 7bx from both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+12x^{2}b+3xb-4b+bx^{4}+7bx^{3}-7bx=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+12b$$

Combine 3xb and -7bx to get -4xb.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+12x^{2}b-4xb-4b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a+12b$$

Subtract 12b from both sides.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+12x^{2}b-4xb-4b+bx^{4}+7bx^{3}-12b=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a$$

Combine -4b and -12b to get -16b.

$$x^{4}+3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a$$

Subtract x^{4} from both sides.

$$3x^{3}-4x^{2}+ax^{3}+3ax^{2}-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-x^{4}$$

Subtract 3x^{3} from both sides.

$$-4x^{2}+ax^{3}+3ax^{2}-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-x^{4}-3x^{3}$$

Add 4x^{2} to both sides.

$$ax^{3}+3ax^{2}-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-x^{4}-3x^{3}+4x^{2}$$

Subtract ax^{3} from both sides.

$$3ax^{2}-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+6x^{3}a+7x^{2}a-6xa-8a-x^{4}-3x^{3}+4x^{2}-ax^{3}$$

Combine 6x^{3}a and -ax^{3} to get 5x^{3}a.

$$3ax^{2}-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+5x^{3}a+7x^{2}a-6xa-8a-x^{4}-3x^{3}+4x^{2}$$

Subtract 3ax^{2} from both sides.

$$-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+5x^{3}a+7x^{2}a-6xa-8a-x^{4}-3x^{3}+4x^{2}-3ax^{2}$$

Combine 7x^{2}a and -3ax^{2} to get 4x^{2}a.

$$-4ax+12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+5x^{3}a+4x^{2}a-6xa-8a-x^{4}-3x^{3}+4x^{2}$$

Add 4ax to both sides.

$$12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+5x^{3}a+4x^{2}a-6xa-8a-x^{4}-3x^{3}+4x^{2}+4ax$$

Combine -6xa and 4ax to get -2xa.

$$12x^{2}b-4xb-16b+bx^{4}+7bx^{3}=x^{4}a+5x^{3}a+4x^{2}a-2xa-8a-x^{4}-3x^{3}+4x^{2}$$

Combine all terms containing b.

$$\left(12x^{2}-4x-16+x^{4}+7x^{3}\right)b=x^{4}a+5x^{3}a+4x^{2}a-2xa-8a-x^{4}-3x^{3}+4x^{2}$$

The equation is in standard form.

$$\left(x^{4}+7x^{3}+12x^{2}-4x-16\right)b=ax^{4}-x^{4}+5ax^{3}-3x^{3}+4ax^{2}+4x^{2}-2ax-8a$$

Divide both sides by 12x^{2}-4x-16+x^{4}+7x^{3}.

$$\frac{\left(x^{4}+7x^{3}+12x^{2}-4x-16\right)b}{x^{4}+7x^{3}+12x^{2}-4x-16}=\frac{\left(x-1\right)\left(x+4\right)\left(ax^{2}-x^{2}+2ax+2a\right)}{x^{4}+7x^{3}+12x^{2}-4x-16}$$

Dividing by 12x^{2}-4x-16+x^{4}+7x^{3} undoes the multiplication by 12x^{2}-4x-16+x^{4}+7x^{3}.

$$b=\frac{\left(x-1\right)\left(x+4\right)\left(ax^{2}-x^{2}+2ax+2a\right)}{x^{4}+7x^{3}+12x^{2}-4x-16}$$

Divide \left(-1+x\right)\left(4+x\right)\left(-x^{2}+2a+2xa+x^{2}a\right) by 12x^{2}-4x-16+x^{4}+7x^{3}.

$$b=\frac{ax^{2}-x^{2}+2ax+2a}{\left(x+2\right)^{2}}$$