Solve -(x-1)^2+xx^2+2x+1+2x=x^2+hx+1

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

Solve for h

$$h=x^{2}-2x+6-\frac{1}{x}$$

$$x\neq 0$$

Steps for Solving Linear Equation

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$-\left(x-1\right)^{2}+x^{3}+2x+1+2x=x^{2}+hx+1$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$-\left(x^{2}-2x+1\right)+x^{3}+2x+1+2x=x^{2}+hx+1$$

To find the opposite of x^{2}-2x+1, find the opposite of each term.

$$-x^{2}+2x-1+x^{3}+2x+1+2x=x^{2}+hx+1$$

Combine 2x and 2x to get 4x.

$$-x^{2}+4x-1+x^{3}+1+2x=x^{2}+hx+1$$

Add -1 and 1 to get 0.

$$-x^{2}+4x+x^{3}+2x=x^{2}+hx+1$$

Combine 4x and 2x to get 6x.

$$-x^{2}+6x+x^{3}=x^{2}+hx+1$$

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

$$x^{2}+hx+1=-x^{2}+6x+x^{3}$$

Subtract x^{2} from both sides.

$$hx+1=-x^{2}+6x+x^{3}-x^{2}$$

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

$$hx+1=-2x^{2}+6x+x^{3}$$

Subtract 1 from both sides.

$$hx=-2x^{2}+6x+x^{3}-1$$

The equation is in standard form.

$$xh=x^{3}-2x^{2}+6x-1$$

Divide both sides by x.

$$\frac{xh}{x}=\frac{x^{3}-2x^{2}+6x-1}{x}$$

Dividing by x undoes the multiplication by x.

$$h=\frac{x^{3}-2x^{2}+6x-1}{x}$$

Divide -2x^{2}+6x-1+x^{3} by x.

$$h=x^{2}-2x+6-\frac{1}{x}$$