Solve (ax+3)(2x+b)=4x^2+10x+6

$$( a x + 3 ) ( 2 x + b ) = 4 x ^ { 2 } + 10 x + 6$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{-4x^{2}-4x+3b-6}{x\left(2x+b\right)}\text{, }&x\neq -\frac{b}{2}\text{ and }x\neq 0\\a\in \mathrm{C}\text{, }&\left(b=2\text{ and }x=0\right)\text{ or }\left(x=-1\text{ and }b=2\right)\text{ or }\left(x=-\frac{3}{2}\text{ and }b=3\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply ax+3 by 2x+b.

$$2ax^{2}+axb+6x+3b=4x^{2}+10x+6$$

Subtract 6x from both sides.

$$2ax^{2}+axb+3b=4x^{2}+10x+6-6x$$

Combine 10x and -6x to get 4x.

$$2ax^{2}+axb+3b=4x^{2}+4x+6$$

Subtract 3b from both sides.

$$2ax^{2}+axb=4x^{2}+4x+6-3b$$

Combine all terms containing a.

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

The equation is in standard form.

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

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

$$\frac{\left(2x^{2}+bx\right)a}{2x^{2}+bx}=\frac{4x^{2}+4x-3b+6}{2x^{2}+bx}$$

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

$$a=\frac{4x^{2}+4x-3b+6}{2x^{2}+bx}$$

Divide 4x^{2}+4x+6-3b by 2x^{2}+bx.

$$a=\frac{4x^{2}+4x-3b+6}{x\left(2x+b\right)}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{2\left(ax^{2}-2x^{2}-2x-3\right)}{ax+3}\text{, }&x=0\text{ or }a\neq -\frac{3}{x}\\b\in \mathrm{C}\text{, }&\left(a=3\text{ and }x=-1\right)\text{ or }\left(a=2\text{ and }x=-\frac{3}{2}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply ax+3 by 2x+b.

$$2ax^{2}+axb+6x+3b=4x^{2}+10x+6$$

Subtract 2ax^{2} from both sides.

$$axb+6x+3b=4x^{2}+10x+6-2ax^{2}$$

Subtract 6x from both sides.

$$axb+3b=4x^{2}+10x+6-2ax^{2}-6x$$

Combine 10x and -6x to get 4x.

$$axb+3b=4x^{2}+4x+6-2ax^{2}$$

Combine all terms containing b.

$$\left(ax+3\right)b=4x^{2}+4x+6-2ax^{2}$$

The equation is in standard form.

$$\left(ax+3\right)b=6+4x+4x^{2}-2ax^{2}$$

Divide both sides by ax+3.

$$\frac{\left(ax+3\right)b}{ax+3}=\frac{6+4x+4x^{2}-2ax^{2}}{ax+3}$$

Dividing by ax+3 undoes the multiplication by ax+3.

$$b=\frac{6+4x+4x^{2}-2ax^{2}}{ax+3}$$

Divide 4x^{2}+4x+6-2ax^{2} by ax+3.

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

Solve for a

$$\left\{\begin{matrix}a=-\frac{-4x^{2}-4x+3b-6}{x\left(2x+b\right)}\text{, }&x\neq -\frac{b}{2}\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&\left(b=2\text{ and }x=0\right)\text{ or }\left(x=-1\text{ and }b=2\right)\text{ or }\left(x=-\frac{3}{2}\text{ and }b=3\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply ax+3 by 2x+b.

$$2ax^{2}+axb+6x+3b=4x^{2}+10x+6$$

Subtract 6x from both sides.

$$2ax^{2}+axb+3b=4x^{2}+10x+6-6x$$

Combine 10x and -6x to get 4x.

$$2ax^{2}+axb+3b=4x^{2}+4x+6$$

Subtract 3b from both sides.

$$2ax^{2}+axb=4x^{2}+4x+6-3b$$

Combine all terms containing a.

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

The equation is in standard form.

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

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

$$\frac{\left(2x^{2}+bx\right)a}{2x^{2}+bx}=\frac{4x^{2}+4x-3b+6}{2x^{2}+bx}$$

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

$$a=\frac{4x^{2}+4x-3b+6}{2x^{2}+bx}$$

Divide 4x^{2}+4x+6-3b by 2x^{2}+bx.

$$a=\frac{4x^{2}+4x-3b+6}{x\left(2x+b\right)}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{2\left(ax^{2}-2x^{2}-2x-3\right)}{ax+3}\text{, }&x=0\text{ or }a\neq -\frac{3}{x}\\b\in \mathrm{R}\text{, }&\left(a=3\text{ and }x=-1\right)\text{ or }\left(a=2\text{ and }x=-\frac{3}{2}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply ax+3 by 2x+b.

$$2ax^{2}+axb+6x+3b=4x^{2}+10x+6$$

Subtract 2ax^{2} from both sides.

$$axb+6x+3b=4x^{2}+10x+6-2ax^{2}$$

Subtract 6x from both sides.

$$axb+3b=4x^{2}+10x+6-2ax^{2}-6x$$

Combine 10x and -6x to get 4x.

$$axb+3b=4x^{2}+4x+6-2ax^{2}$$

Combine all terms containing b.

$$\left(ax+3\right)b=4x^{2}+4x+6-2ax^{2}$$

The equation is in standard form.

$$\left(ax+3\right)b=6+4x+4x^{2}-2ax^{2}$$

Divide both sides by ax+3.

$$\frac{\left(ax+3\right)b}{ax+3}=\frac{6+4x+4x^{2}-2ax^{2}}{ax+3}$$

Dividing by ax+3 undoes the multiplication by ax+3.

$$b=\frac{6+4x+4x^{2}-2ax^{2}}{ax+3}$$

Divide 4x^{2}+4x+6-2ax^{2} by ax+3.

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