Solve x^2+hx+2x+2x/6+x=x

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

Solve for h (complex solution)

$$\left\{\begin{matrix}h=2-5x-x^{2}\text{, }&x\neq -6\\h\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x+6.

$$\left(x+6\right)x^{2}+hx+2x+2x=x\left(x+6\right)$$

Use the distributive property to multiply x+6 by x^{2}.

$$x^{3}+6x^{2}+hx+2x+2x=x\left(x+6\right)$$

Combine 2x and 2x to get 4x.

$$x^{3}+6x^{2}+hx+4x=x\left(x+6\right)$$

Use the distributive property to multiply x by x+6.

$$x^{3}+6x^{2}+hx+4x=x^{2}+6x$$

Subtract x^{3} from both sides.

$$6x^{2}+hx+4x=x^{2}+6x-x^{3}$$

Subtract 6x^{2} from both sides.

$$hx+4x=x^{2}+6x-x^{3}-6x^{2}$$

Combine x^{2} and -6x^{2} to get -5x^{2}.

$$hx+4x=-5x^{2}+6x-x^{3}$$

Subtract 4x from both sides.

$$hx=-5x^{2}+6x-x^{3}-4x$$

Combine 6x and -4x to get 2x.

$$hx=-5x^{2}+2x-x^{3}$$

The equation is in standard form.

$$xh=2x-5x^{2}-x^{3}$$

Divide both sides by x.

$$\frac{xh}{x}=\frac{x\left(2-5x-x^{2}\right)}{x}$$

Dividing by x undoes the multiplication by x.

$$h=\frac{x\left(2-5x-x^{2}\right)}{x}$$

Divide x\left(-5x+2-x^{2}\right) by x.

$$h=2-5x-x^{2}$$

Solve for h

$$\left\{\begin{matrix}h=2-5x-x^{2}\text{, }&x\neq -6\\h\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x+6.

$$\left(x+6\right)x^{2}+hx+2x+2x=x\left(x+6\right)$$

Use the distributive property to multiply x+6 by x^{2}.

$$x^{3}+6x^{2}+hx+2x+2x=x\left(x+6\right)$$

Combine 2x and 2x to get 4x.

$$x^{3}+6x^{2}+hx+4x=x\left(x+6\right)$$

Use the distributive property to multiply x by x+6.

$$x^{3}+6x^{2}+hx+4x=x^{2}+6x$$

Subtract x^{3} from both sides.

$$6x^{2}+hx+4x=x^{2}+6x-x^{3}$$

Subtract 6x^{2} from both sides.

$$hx+4x=x^{2}+6x-x^{3}-6x^{2}$$

Combine x^{2} and -6x^{2} to get -5x^{2}.

$$hx+4x=-5x^{2}+6x-x^{3}$$

Subtract 4x from both sides.

$$hx=-5x^{2}+6x-x^{3}-4x$$

Combine 6x and -4x to get 2x.

$$hx=-5x^{2}+2x-x^{3}$$

The equation is in standard form.

$$xh=2x-5x^{2}-x^{3}$$

Divide both sides by x.

$$\frac{xh}{x}=\frac{x\left(2-5x-x^{2}\right)}{x}$$

Dividing by x undoes the multiplication by x.

$$h=\frac{x\left(2-5x-x^{2}\right)}{x}$$

Divide x\left(-5x+2-x^{2}\right) by x.

$$h=2-5x-x^{2}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{\sqrt{33-4h}-5}{2}\text{; }x=0\text{, }&\text{unconditionally}\\x=\frac{-\sqrt{33-4h}-5}{2}\text{, }&h\neq -4\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\frac{-\sqrt{33-4h}-5}{2}\text{, }&h\neq -4\text{ and }h\leq \frac{33}{4}\\x=\frac{\sqrt{33-4h}-5}{2}\text{, }&h\leq \frac{33}{4}\end{matrix}\right.$$