Solve (9x-1)(2x+3)=-3+12x-(2a-5b)x+(4a+b)x^2

$$( 9 x - 1 ) ( 2 x + 3 ) = - 3 + 12 x - ( 2 a - 5 b ) x + ( 4 a + b ) x ^ { 2 }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{bx-18x+5b-13}{2\left(2x-1\right)}\text{, }&x\neq \frac{1}{2}\\a\in \mathrm{C}\text{, }&x=0\text{ or }\left(b=4\text{ and }x=\frac{1}{2}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 9x-1 by 2x+3 and combine like terms.

$$18x^{2}+25x-3=-3+12x-\left(2a-5b\right)x+\left(4a+b\right)x^{2}$$

Use the distributive property to multiply 2a-5b by x.

$$18x^{2}+25x-3=-3+12x-\left(2ax-5bx\right)+\left(4a+b\right)x^{2}$$

To find the opposite of 2ax-5bx, find the opposite of each term.

$$18x^{2}+25x-3=-3+12x-2ax+5bx+\left(4a+b\right)x^{2}$$

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

$$18x^{2}+25x-3=-3+12x-2ax+5bx+4ax^{2}+bx^{2}$$

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

$$-3+12x-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x-3$$

Add 3 to both sides.

$$12x-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x-3+3$$

Add -3 and 3 to get 0.

$$12x-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x$$

Subtract 12x from both sides.

$$-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x-12x$$

Combine 25x and -12x to get 13x.

$$-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+13x$$

Subtract 5bx from both sides.

$$-2ax+4ax^{2}+bx^{2}=18x^{2}+13x-5bx$$

Subtract bx^{2} from both sides.

$$-2ax+4ax^{2}=18x^{2}+13x-5bx-bx^{2}$$

Reorder the terms.

$$4ax^{2}-2ax=-bx^{2}+18x^{2}-5bx+13x$$

Combine all terms containing a.

$$\left(4x^{2}-2x\right)a=-bx^{2}+18x^{2}-5bx+13x$$

The equation is in standard form.

$$\left(4x^{2}-2x\right)a=13x-5bx+18x^{2}-bx^{2}$$

Divide both sides by 4x^{2}-2x.

$$\frac{\left(4x^{2}-2x\right)a}{4x^{2}-2x}=\frac{x\left(13-5b+18x-bx\right)}{4x^{2}-2x}$$

Dividing by 4x^{2}-2x undoes the multiplication by 4x^{2}-2x.

$$a=\frac{x\left(13-5b+18x-bx\right)}{4x^{2}-2x}$$

Divide x\left(-bx+18x-5b+13\right) by 4x^{2}-2x.

$$a=\frac{13-5b+18x-bx}{2\left(2x-1\right)}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{4ax-18x-2a-13}{x+5}\text{, }&x\neq -5\\b\in \mathrm{C}\text{, }&x=0\text{ or }\left(a=\frac{7}{2}\text{ and }x=-5\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 9x-1 by 2x+3 and combine like terms.

$$18x^{2}+25x-3=-3+12x-\left(2a-5b\right)x+\left(4a+b\right)x^{2}$$

Use the distributive property to multiply 2a-5b by x.

$$18x^{2}+25x-3=-3+12x-\left(2ax-5bx\right)+\left(4a+b\right)x^{2}$$

To find the opposite of 2ax-5bx, find the opposite of each term.

$$18x^{2}+25x-3=-3+12x-2ax+5bx+\left(4a+b\right)x^{2}$$

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

$$18x^{2}+25x-3=-3+12x-2ax+5bx+4ax^{2}+bx^{2}$$

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

$$-3+12x-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x-3$$

Add 3 to both sides.

$$12x-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x-3+3$$

Add -3 and 3 to get 0.

$$12x-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x$$

Subtract 12x from both sides.

$$-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x-12x$$

Combine 25x and -12x to get 13x.

$$-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+13x$$

Add 2ax to both sides.

$$5bx+4ax^{2}+bx^{2}=18x^{2}+13x+2ax$$

Subtract 4ax^{2} from both sides.

$$5bx+bx^{2}=18x^{2}+13x+2ax-4ax^{2}$$

Combine all terms containing b.

$$\left(5x+x^{2}\right)b=18x^{2}+13x+2ax-4ax^{2}$$

The equation is in standard form.

$$\left(x^{2}+5x\right)b=13x+2ax+18x^{2}-4ax^{2}$$

Divide both sides by 5x+x^{2}.

$$\frac{\left(x^{2}+5x\right)b}{x^{2}+5x}=\frac{x\left(13+2a+18x-4ax\right)}{x^{2}+5x}$$

Dividing by 5x+x^{2} undoes the multiplication by 5x+x^{2}.

$$b=\frac{x\left(13+2a+18x-4ax\right)}{x^{2}+5x}$$

Divide x\left(18x+13+2a-4ax\right) by 5x+x^{2}.

$$b=\frac{13+2a+18x-4ax}{x+5}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{bx-18x+5b-13}{2\left(2x-1\right)}\text{, }&x\neq \frac{1}{2}\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&\left(b=4\text{ and }x=\frac{1}{2}\right)\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 9x-1 by 2x+3 and combine like terms.

$$18x^{2}+25x-3=-3+12x-\left(2a-5b\right)x+\left(4a+b\right)x^{2}$$

Use the distributive property to multiply 2a-5b by x.

$$18x^{2}+25x-3=-3+12x-\left(2ax-5bx\right)+\left(4a+b\right)x^{2}$$

To find the opposite of 2ax-5bx, find the opposite of each term.

$$18x^{2}+25x-3=-3+12x-2ax+5bx+\left(4a+b\right)x^{2}$$

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

$$18x^{2}+25x-3=-3+12x-2ax+5bx+4ax^{2}+bx^{2}$$

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

$$-3+12x-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x-3$$

Add 3 to both sides.

$$12x-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x-3+3$$

Add -3 and 3 to get 0.

$$12x-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x$$

Subtract 12x from both sides.

$$-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x-12x$$

Combine 25x and -12x to get 13x.

$$-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+13x$$

Subtract 5bx from both sides.

$$-2ax+4ax^{2}+bx^{2}=18x^{2}+13x-5bx$$

Subtract bx^{2} from both sides.

$$-2ax+4ax^{2}=18x^{2}+13x-5bx-bx^{2}$$

Reorder the terms.

$$4ax^{2}-2ax=-bx^{2}+18x^{2}-5bx+13x$$

Combine all terms containing a.

$$\left(4x^{2}-2x\right)a=-bx^{2}+18x^{2}-5bx+13x$$

The equation is in standard form.

$$\left(4x^{2}-2x\right)a=13x-5bx+18x^{2}-bx^{2}$$

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

$$\frac{\left(4x^{2}-2x\right)a}{4x^{2}-2x}=\frac{x\left(13-5b+18x-bx\right)}{4x^{2}-2x}$$

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

$$a=\frac{x\left(13-5b+18x-bx\right)}{4x^{2}-2x}$$

Divide x\left(-bx+18x-5b+13\right) by -2x+4x^{2}.

$$a=\frac{13-5b+18x-bx}{2\left(2x-1\right)}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{4ax-18x-2a-13}{x+5}\text{, }&x\neq -5\text{ and }x\neq 0\\b\in \mathrm{R}\text{, }&\left(a=\frac{7}{2}\text{ and }x=-5\right)\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 9x-1 by 2x+3 and combine like terms.

$$18x^{2}+25x-3=-3+12x-\left(2a-5b\right)x+\left(4a+b\right)x^{2}$$

Use the distributive property to multiply 2a-5b by x.

$$18x^{2}+25x-3=-3+12x-\left(2ax-5bx\right)+\left(4a+b\right)x^{2}$$

To find the opposite of 2ax-5bx, find the opposite of each term.

$$18x^{2}+25x-3=-3+12x-2ax+5bx+\left(4a+b\right)x^{2}$$

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

$$18x^{2}+25x-3=-3+12x-2ax+5bx+4ax^{2}+bx^{2}$$

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

$$-3+12x-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x-3$$

Add 3 to both sides.

$$12x-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x-3+3$$

Add -3 and 3 to get 0.

$$12x-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x$$

Subtract 12x from both sides.

$$-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+25x-12x$$

Combine 25x and -12x to get 13x.

$$-2ax+5bx+4ax^{2}+bx^{2}=18x^{2}+13x$$

Add 2ax to both sides.

$$5bx+4ax^{2}+bx^{2}=18x^{2}+13x+2ax$$

Subtract 4ax^{2} from both sides.

$$5bx+bx^{2}=18x^{2}+13x+2ax-4ax^{2}$$

Combine all terms containing b.

$$\left(5x+x^{2}\right)b=18x^{2}+13x+2ax-4ax^{2}$$

The equation is in standard form.

$$\left(x^{2}+5x\right)b=13x+2ax+18x^{2}-4ax^{2}$$

Divide both sides by x^{2}+5x.

$$\frac{\left(x^{2}+5x\right)b}{x^{2}+5x}=\frac{x\left(13+2a+18x-4ax\right)}{x^{2}+5x}$$

Dividing by x^{2}+5x undoes the multiplication by x^{2}+5x.

$$b=\frac{x\left(13+2a+18x-4ax\right)}{x^{2}+5x}$$

Divide x\left(18x+13+2a-4ax\right) by x^{2}+5x.

$$b=\frac{13+2a+18x-4ax}{x+5}$$