Solve x^2+ax+1=(x+2)^2-b

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

Solve for a (complex solution)

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

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

$$x^{2}+ax+1=x^{2}+4x+4-b$$

Subtract x^{2} from both sides.

$$ax+1=x^{2}+4x+4-b-x^{2}$$

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

$$ax+1=4x+4-b$$

Subtract 1 from both sides.

$$ax=4x+4-b-1$$

Subtract 1 from 4 to get 3.

$$ax=4x+3-b$$

The equation is in standard form.

$$xa=4x-b+3$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{4x-b+3}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{4x-b+3}{x}$$

Solve for a

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

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

$$x^{2}+ax+1=x^{2}+4x+4-b$$

Subtract x^{2} from both sides.

$$ax+1=x^{2}+4x+4-b-x^{2}$$

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

$$ax+1=4x+4-b$$

Subtract 1 from both sides.

$$ax=4x+4-b-1$$

Subtract 1 from 4 to get 3.

$$ax=4x+3-b$$

The equation is in standard form.

$$xa=4x-b+3$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{4x-b+3}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{4x-b+3}{x}$$

Solve for b

$$b=3+4x-ax$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

$$x^{2}+ax+1=x^{2}+4x+4-b$$

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

$$x^{2}+4x+4-b=x^{2}+ax+1$$

Subtract x^{2} from both sides.

$$4x+4-b=x^{2}+ax+1-x^{2}$$

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

$$4x+4-b=ax+1$$

Subtract 4x from both sides.

$$4-b=ax+1-4x$$

Subtract 4 from both sides.

$$-b=ax+1-4x-4$$

Subtract 4 from 1 to get -3.

$$-b=ax-3-4x$$

The equation is in standard form.

$$-b=ax-4x-3$$

Divide both sides by -1.

$$\frac{-b}{-1}=\frac{ax-4x-3}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$b=\frac{ax-4x-3}{-1}$$

Divide ax-3-4x by -1.

$$b=3+4x-ax$$