Solve H(x)=f(x)/g(x)=x^2+mx/e^x

$$H ( x ) = \frac { f ( x ) } { g ( x ) } = \frac { x ^ { 2 } + m x } { e ^ { x } }$$

Solve for f

$$f=\frac{gx\left(x+m\right)}{e^{x}}$$

$$g\neq 0\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by gx.

$$fx=gxe^{-x}\left(x^{2}+mx\right)$$

Use the distributive property to multiply gxe^{-x} by x^{2}+mx.

$$fx=ge^{-x}x^{3}+ge^{-x}mx^{2}$$

The equation is in standard form.

$$xf=\frac{gx^{3}+gmx^{2}}{e^{x}}$$

Divide both sides by x.

$$\frac{xf}{x}=\frac{g\left(x+m\right)x^{2}}{e^{x}x}$$

Dividing by x undoes the multiplication by x.

$$f=\frac{g\left(x+m\right)x^{2}}{e^{x}x}$$

Divide \frac{g\left(x+m\right)x^{2}}{e^{x}} by x.

$$f=\frac{gx\left(x+m\right)}{e^{x}}$$

Solve for g

$$\left\{\begin{matrix}g=\frac{fe^{x}}{x\left(x+m\right)}\text{, }&f\neq 0\text{ and }x\neq -m\text{ and }x\neq 0\\g\neq 0\text{, }&m=-x\text{ and }f=0\text{ and }x\neq 0\end{matrix}\right.$$

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.

$$fx=gxe^{-x}\left(x^{2}+mx\right)$$

Use the distributive property to multiply gxe^{-x} by x^{2}+mx.

$$fx=ge^{-x}x^{3}+ge^{-x}mx^{2}$$

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

$$ge^{-x}x^{3}+ge^{-x}mx^{2}=fx$$

Combine all terms containing g.

$$\left(e^{-x}x^{3}+e^{-x}mx^{2}\right)g=fx$$

The equation is in standard form.

$$\frac{x^{3}+mx^{2}}{e^{x}}g=fx$$

Divide both sides by e^{-x}x^{3}+e^{-x}mx^{2}.

$$\frac{\frac{x^{3}+mx^{2}}{e^{x}}ge^{x}}{x^{3}+mx^{2}}=\frac{fxe^{x}}{x^{3}+mx^{2}}$$

Dividing by e^{-x}x^{3}+e^{-x}mx^{2} undoes the multiplication by e^{-x}x^{3}+e^{-x}mx^{2}.

$$g=\frac{fxe^{x}}{x^{3}+mx^{2}}$$

Divide fx by e^{-x}x^{3}+e^{-x}mx^{2}.

$$g=\frac{fe^{x}}{x\left(x+m\right)}$$

Variable g cannot be equal to 0.

$$g=\frac{fe^{x}}{x\left(x+m\right)}\text{, }g\neq 0$$