Solve x^2+2x(a+1)=(x-a)^2+2(x+a)

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

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{a+2}{4}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&a=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x by a+1.

$$x^{2}+2xa+2x=\left(x-a\right)^{2}+2\left(x+a\right)$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(x-a\right)^{2}.

$$x^{2}+2xa+2x=x^{2}-2xa+a^{2}+2\left(x+a\right)$$

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

$$x^{2}+2xa+2x=x^{2}-2xa+a^{2}+2x+2a$$

Subtract x^{2} from both sides.

$$x^{2}+2xa+2x-x^{2}=-2xa+a^{2}+2x+2a$$

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

$$2xa+2x=-2xa+a^{2}+2x+2a$$

Add 2xa to both sides.

$$2xa+2x+2xa=a^{2}+2x+2a$$

Combine 2xa and 2xa to get 4xa.

$$4xa+2x=a^{2}+2x+2a$$

Subtract 2x from both sides.

$$4xa+2x-2x=a^{2}+2a$$

Combine 2x and -2x to get 0.

$$4xa=a^{2}+2a$$

The equation is in standard form.

$$4ax=a^{2}+2a$$

Divide both sides by 4a.

$$\frac{4ax}{4a}=\frac{a\left(a+2\right)}{4a}$$

Dividing by 4a undoes the multiplication by 4a.

$$x=\frac{a\left(a+2\right)}{4a}$$

Divide a\left(2+a\right) by 4a.

$$x=\frac{a}{4}+\frac{1}{2}$$

Solve for x

$$\left\{\begin{matrix}\\x=\frac{a+2}{4}\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&a=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x by a+1.

$$x^{2}+2xa+2x=\left(x-a\right)^{2}+2\left(x+a\right)$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(x-a\right)^{2}.

$$x^{2}+2xa+2x=x^{2}-2xa+a^{2}+2\left(x+a\right)$$

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

$$x^{2}+2xa+2x=x^{2}-2xa+a^{2}+2x+2a$$

Subtract x^{2} from both sides.

$$x^{2}+2xa+2x-x^{2}=-2xa+a^{2}+2x+2a$$

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

$$2xa+2x=-2xa+a^{2}+2x+2a$$

Add 2xa to both sides.

$$2xa+2x+2xa=a^{2}+2x+2a$$

Combine 2xa and 2xa to get 4xa.

$$4xa+2x=a^{2}+2x+2a$$

Subtract 2x from both sides.

$$4xa+2x-2x=a^{2}+2a$$

Combine 2x and -2x to get 0.

$$4xa=a^{2}+2a$$

The equation is in standard form.

$$4ax=a^{2}+2a$$

Divide both sides by 4a.

$$\frac{4ax}{4a}=\frac{a\left(a+2\right)}{4a}$$

Dividing by 4a undoes the multiplication by 4a.

$$x=\frac{a\left(a+2\right)}{4a}$$

Divide a\left(2+a\right) by 4a.

$$x=\frac{a}{4}+\frac{1}{2}$$

Solve for a

$$a=4x-2$$

$$a=0$$