Solve x^2+n=(x-1)(x+m)+2

$$x ^ { 2 } + n = ( x - 1 ) ( x + m ) + 2$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=-\frac{2-n-x}{x-1}\text{, }&x\neq 1\\m\in \mathrm{C}\text{, }&n=1\text{ and }x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-1 by x+m.

$$x^{2}+n=x^{2}+xm-x-m+2$$

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

$$x^{2}+xm-x-m+2=x^{2}+n$$

Subtract x^{2} from both sides.

$$xm-x-m+2=x^{2}+n-x^{2}$$

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

$$xm-x-m+2=n$$

Add x to both sides.

$$xm-m+2=n+x$$

Subtract 2 from both sides.

$$xm-m=n+x-2$$

Combine all terms containing m.

$$\left(x-1\right)m=n+x-2$$

The equation is in standard form.

$$\left(x-1\right)m=x+n-2$$

Divide both sides by x-1.

$$\frac{\left(x-1\right)m}{x-1}=\frac{x+n-2}{x-1}$$

Dividing by x-1 undoes the multiplication by x-1.

$$m=\frac{x+n-2}{x-1}$$

Solve for m

$$\left\{\begin{matrix}m=-\frac{2-n-x}{x-1}\text{, }&x\neq 1\\m\in \mathrm{R}\text{, }&n=1\text{ and }x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-1 by x+m.

$$x^{2}+n=x^{2}+xm-x-m+2$$

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

$$x^{2}+xm-x-m+2=x^{2}+n$$

Subtract x^{2} from both sides.

$$xm-x-m+2=x^{2}+n-x^{2}$$

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

$$xm-x-m+2=n$$

Add x to both sides.

$$xm-m+2=n+x$$

Subtract 2 from both sides.

$$xm-m=n+x-2$$

Combine all terms containing m.

$$\left(x-1\right)m=n+x-2$$

The equation is in standard form.

$$\left(x-1\right)m=x+n-2$$

Divide both sides by x-1.

$$\frac{\left(x-1\right)m}{x-1}=\frac{x+n-2}{x-1}$$

Dividing by x-1 undoes the multiplication by x-1.

$$m=\frac{x+n-2}{x-1}$$

Solve for n

$$n=mx-x-m+2$$

Solution Steps

Use the distributive property to multiply x-1 by x+m.

$$x^{2}+n=x^{2}+xm-x-m+2$$

Subtract x^{2} from both sides.

$$n=x^{2}+xm-x-m+2-x^{2}$$

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

$$n=xm-x-m+2$$