Solve -(x)/g(x)=4-3x-x^2/x-2

$$\frac { - ( x ) } { g ( x ) } = \frac { 4 - 3 x - x ^ { 2 } } { x - 2 }$$

Solve for g

$$g=-\frac{2-x}{\left(x-1\right)\left(x+4\right)}$$

$$x\neq 2\text{ and }x\neq -4\text{ and }x\neq 1\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Variable g cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by gx\left(x-2\right), the least common multiple of gx,x-2.

$$\left(x-2\right)\left(-x\right)=gx\left(4-3x-x^{2}\right)$$

Use the distributive property to multiply x-2 by -x.

$$x\left(-x\right)-2\left(-x\right)=gx\left(4-3x-x^{2}\right)$$

Multiply -2 and -1 to get 2.

$$x\left(-x\right)+2x=gx\left(4-3x-x^{2}\right)$$

Use the distributive property to multiply gx by 4-3x-x^{2}.

$$x\left(-x\right)+2x=4gx-3gx^{2}-gx^{3}$$

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

$$4gx-3gx^{2}-gx^{3}=x\left(-x\right)+2x$$

Multiply x and x to get x^{2}.

$$4gx-3gx^{2}-gx^{3}=x^{2}\left(-1\right)+2x$$

Combine all terms containing g.

$$\left(4x-3x^{2}-x^{3}\right)g=x^{2}\left(-1\right)+2x$$

The equation is in standard form.

$$\left(4x-3x^{2}-x^{3}\right)g=2x-x^{2}$$

Divide both sides by 4x-3x^{2}-x^{3}.

$$\frac{\left(4x-3x^{2}-x^{3}\right)g}{4x-3x^{2}-x^{3}}=\frac{x\left(2-x\right)}{4x-3x^{2}-x^{3}}$$

Dividing by 4x-3x^{2}-x^{3} undoes the multiplication by 4x-3x^{2}-x^{3}.

$$g=\frac{x\left(2-x\right)}{4x-3x^{2}-x^{3}}$$

Divide x\left(2-x\right) by 4x-3x^{2}-x^{3}.

$$g=\frac{2-x}{\left(1-x\right)\left(x+4\right)}$$

Variable g cannot be equal to 0.

$$g=\frac{2-x}{\left(1-x\right)\left(x+4\right)}\text{, }g\neq 0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{25g^{2}-14g+1}-3g+1}{2g}\text{, }&g\neq 0\\x=\frac{\sqrt{25g^{2}-14g+1}-3g+1}{2g}\text{, }&g\neq 0\text{ and }g\neq \frac{1}{2}\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-\sqrt{25g^{2}-14g+1}-3g+1}{2g}\text{, }&\left(g\neq 0\text{ and }g\leq \frac{7-2\sqrt{6}}{25}\right)\text{ or }g\geq \frac{2\sqrt{6}+7}{25}\\x=\frac{\sqrt{25g^{2}-14g+1}-3g+1}{2g}\text{, }&\left(g\neq 0\text{ and }g\leq \frac{7-2\sqrt{6}}{25}\right)\text{ or }\left(g\neq \frac{1}{2}\text{ and }g\geq \frac{2\sqrt{6}+7}{25}\right)\end{matrix}\right.$$