Solve 1x^3+ax^2+18x+b=(x^2-2x+3)*(4x+a+8)+8x+16+2a

$$1 x ^ { 3 } + a x ^ { 2 } + 18 x + b = ( x ^ { 2 } - 2 x + 3 ) \cdot ( 4 x + a + 8 ) + 8 x + 16 + 2 a$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{-3x^{3}+14x+b-40}{2x-5}\text{, }&x\neq \frac{5}{2}\\a\in \mathrm{R}\text{, }&b=\frac{415}{8}\text{ and }x=\frac{5}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{2}-2x+3 by 4x+a+8 and combine like terms.

$$1x^{3}+ax^{2}+18x+b=4x^{3}+x^{2}a-2xa-4x+3a+24+8x+16+2a$$

Combine -4x and 8x to get 4x.

$$1x^{3}+ax^{2}+18x+b=4x^{3}+x^{2}a-2xa+4x+3a+24+16+2a$$

Add 24 and 16 to get 40.

$$1x^{3}+ax^{2}+18x+b=4x^{3}+x^{2}a-2xa+4x+3a+40+2a$$

Combine 3a and 2a to get 5a.

$$1x^{3}+ax^{2}+18x+b=4x^{3}+x^{2}a-2xa+4x+5a+40$$

Subtract x^{2}a from both sides.

$$1x^{3}+ax^{2}+18x+b-x^{2}a=4x^{3}-2xa+4x+5a+40$$

Combine ax^{2} and -x^{2}a to get 0.

$$1x^{3}+18x+b=4x^{3}-2xa+4x+5a+40$$

Swap sides so that all variable terms are on the left hand side.

$$4x^{3}-2xa+4x+5a+40=1x^{3}+18x+b$$

Subtract 4x^{3} from both sides.

$$-2xa+4x+5a+40=1x^{3}+18x+b-4x^{3}$$

Combine 1x^{3} and -4x^{3} to get -3x^{3}.

$$-2xa+4x+5a+40=-3x^{3}+18x+b$$

Subtract 4x from both sides.

$$-2xa+5a+40=-3x^{3}+18x+b-4x$$

Combine 18x and -4x to get 14x.

$$-2xa+5a+40=-3x^{3}+14x+b$$

Subtract 40 from both sides.

$$-2xa+5a=-3x^{3}+14x+b-40$$

Combine all terms containing a.

$$\left(-2x+5\right)a=-3x^{3}+14x+b-40$$

The equation is in standard form.

$$\left(5-2x\right)a=-3x^{3}+14x+b-40$$

Divide both sides by -2x+5.

$$\frac{\left(5-2x\right)a}{5-2x}=\frac{-3x^{3}+14x+b-40}{5-2x}$$

Dividing by -2x+5 undoes the multiplication by -2x+5.

$$a=\frac{-3x^{3}+14x+b-40}{5-2x}$$

Solve for b

$$b=3x^{3}-2ax-14x+5a+40$$

Solution Steps

Use the distributive property to multiply x^{2}-2x+3 by 4x+a+8 and combine like terms.

$$1x^{3}+ax^{2}+18x+b=4x^{3}+x^{2}a-2xa-4x+3a+24+8x+16+2a$$

Combine -4x and 8x to get 4x.

$$1x^{3}+ax^{2}+18x+b=4x^{3}+x^{2}a-2xa+4x+3a+24+16+2a$$

Add 24 and 16 to get 40.

$$1x^{3}+ax^{2}+18x+b=4x^{3}+x^{2}a-2xa+4x+3a+40+2a$$

Combine 3a and 2a to get 5a.

$$1x^{3}+ax^{2}+18x+b=4x^{3}+x^{2}a-2xa+4x+5a+40$$

Subtract 1x^{3} from both sides.

$$ax^{2}+18x+b=4x^{3}+x^{2}a-2xa+4x+5a+40-x^{3}$$

Combine 4x^{3} and -x^{3} to get 3x^{3}.

$$ax^{2}+18x+b=3x^{3}+x^{2}a-2xa+4x+5a+40$$

Subtract ax^{2} from both sides.

$$18x+b=3x^{3}+x^{2}a-2xa+4x+5a+40-ax^{2}$$

Combine x^{2}a and -ax^{2} to get 0.

$$18x+b=3x^{3}-2xa+4x+5a+40$$

Subtract 18x from both sides.

$$b=3x^{3}-2xa+4x+5a+40-18x$$

Combine 4x and -18x to get -14x.

$$b=3x^{3}-2xa-14x+5a+40$$