Solve (2x+a)(bx-4)=-2x^2+cx+12

$$( 2 x + a ) ( b x - 4 ) = - 2 x ^ { 2 } + c x + 12$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{2bx^{2}+2x^{2}-cx-8x-12}{bx-4}\text{, }&b=0\text{ or }x\neq \frac{4}{b}\\a\in \mathrm{R}\text{, }&c=-3b+\frac{8}{b}\text{ and }x=\frac{4}{b}\text{ and }b\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x+a by bx-4.

$$2bx^{2}-8x+abx-4a=-2x^{2}+cx+12$$

Subtract 2bx^{2} from both sides.

$$-8x+abx-4a=-2x^{2}+cx+12-2bx^{2}$$

Add 8x to both sides.

$$abx-4a=-2x^{2}+cx+12-2bx^{2}+8x$$

Combine all terms containing a.

$$\left(bx-4\right)a=-2x^{2}+cx+12-2bx^{2}+8x$$

The equation is in standard form.

$$\left(bx-4\right)a=12+8x+cx-2x^{2}-2bx^{2}$$

Divide both sides by bx-4.

$$\frac{\left(bx-4\right)a}{bx-4}=\frac{12+8x+cx-2x^{2}-2bx^{2}}{bx-4}$$

Dividing by bx-4 undoes the multiplication by bx-4.

$$a=\frac{12+8x+cx-2x^{2}-2bx^{2}}{bx-4}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{2x^{2}-cx-8x-4a-12}{x\left(2x+a\right)}\text{, }&x\neq -\frac{a}{2}\text{ and }x\neq 0\\b\in \mathrm{R}\text{, }&\left(x=\frac{-\sqrt{c^{2}+96}+c}{4}\text{ and }a=\frac{\sqrt{c^{2}+96}-c}{2}\right)\text{ or }\left(x=\frac{\sqrt{c^{2}+96}+c}{4}\text{ and }a=\frac{-\sqrt{c^{2}+96}-c}{2}\right)\text{ or }\left(a=-3\text{ and }x=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x+a by bx-4.

$$2bx^{2}-8x+abx-4a=-2x^{2}+cx+12$$

Add 8x to both sides.

$$2bx^{2}+abx-4a=-2x^{2}+cx+12+8x$$

Add 4a to both sides.

$$2bx^{2}+abx=-2x^{2}+cx+12+8x+4a$$

Combine all terms containing b.

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

The equation is in standard form.

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

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

$$\frac{\left(2x^{2}+ax\right)b}{2x^{2}+ax}=\frac{12+4a+8x+cx-2x^{2}}{2x^{2}+ax}$$

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

$$b=\frac{12+4a+8x+cx-2x^{2}}{2x^{2}+ax}$$

Divide -2x^{2}+cx+12+8x+4a by 2x^{2}+ax.

$$b=\frac{12+4a+8x+cx-2x^{2}}{x\left(2x+a\right)}$$