Solve mx^2+(2m+n)x+2n=0

$$m x ^ { 2 } + ( 2 m + n ) x + 2 n = 0$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=-\frac{n}{x}\text{, }&x\neq 0\\m\in \mathrm{C}\text{, }&x=-2\text{ or }\left(n=0\text{ and }x=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2m+n by x.

$$mx^{2}+2mx+nx+2n=0$$

Subtract nx from both sides. Anything subtracted from zero gives its negation.

$$mx^{2}+2mx+2n=-nx$$

Subtract 2n from both sides.

$$mx^{2}+2mx=-nx-2n$$

Combine all terms containing m.

$$\left(x^{2}+2x\right)m=-nx-2n$$

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

$$\frac{\left(x^{2}+2x\right)m}{x^{2}+2x}=-\frac{n\left(x+2\right)}{x^{2}+2x}$$

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

$$m=-\frac{n\left(x+2\right)}{x^{2}+2x}$$

Divide -n\left(2+x\right) by x^{2}+2x.

$$m=-\frac{n}{x}$$

Solve for n (complex solution)

$$\left\{\begin{matrix}\\n=-mx\text{, }&\text{unconditionally}\\n\in \mathrm{C}\text{, }&x=-2\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2m+n by x.

$$mx^{2}+2mx+nx+2n=0$$

Subtract mx^{2} from both sides. Anything subtracted from zero gives its negation.

$$2mx+nx+2n=-mx^{2}$$

Subtract 2mx from both sides.

$$nx+2n=-mx^{2}-2mx$$

Combine all terms containing n.

$$\left(x+2\right)n=-mx^{2}-2mx$$

Divide both sides by 2+x.

$$\frac{\left(x+2\right)n}{x+2}=-\frac{mx\left(x+2\right)}{x+2}$$

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

$$n=-\frac{mx\left(x+2\right)}{x+2}$$

Divide -mx\left(2+x\right) by 2+x.

$$n=-mx$$

Solve for m

$$\left\{\begin{matrix}m=-\frac{n}{x}\text{, }&x\neq 0\\m\in \mathrm{R}\text{, }&x=-2\text{ or }\left(n=0\text{ and }x=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2m+n by x.

$$mx^{2}+2mx+nx+2n=0$$

Subtract nx from both sides. Anything subtracted from zero gives its negation.

$$mx^{2}+2mx+2n=-nx$$

Subtract 2n from both sides.

$$mx^{2}+2mx=-nx-2n$$

Combine all terms containing m.

$$\left(x^{2}+2x\right)m=-nx-2n$$

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

$$\frac{\left(x^{2}+2x\right)m}{x^{2}+2x}=-\frac{n\left(x+2\right)}{x^{2}+2x}$$

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

$$m=-\frac{n\left(x+2\right)}{x^{2}+2x}$$

Divide -n\left(2+x\right) by x^{2}+2x.

$$m=-\frac{n}{x}$$

Solve for n

$$\left\{\begin{matrix}\\n=-mx\text{, }&\text{unconditionally}\\n\in \mathrm{R}\text{, }&x=-2\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2m+n by x.

$$mx^{2}+2mx+nx+2n=0$$

Subtract mx^{2} from both sides. Anything subtracted from zero gives its negation.

$$2mx+nx+2n=-mx^{2}$$

Subtract 2mx from both sides.

$$nx+2n=-mx^{2}-2mx$$

Combine all terms containing n.

$$\left(x+2\right)n=-mx^{2}-2mx$$

Divide both sides by 2+x.

$$\frac{\left(x+2\right)n}{x+2}=-\frac{mx\left(x+2\right)}{x+2}$$

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

$$n=-\frac{mx\left(x+2\right)}{x+2}$$

Divide -mx\left(2+x\right) by 2+x.

$$n=-mx$$