Question

For the following exercises, find $R\left(x\right)=f\left(x\right)g\left(x\right)$where $f\left(x\right)andg\left(x\right)$are given.

$f\left(x\right)=\frac{2x^2+8x}{x^2-9x+20}andg\left(x\right)=\frac{x-5}{x^2}$

Short answer

The value of$R\left(x\right)=\frac{2\left(x+4\right)}{x\left(x-4\right)}.$

Step-by-step solution

Step 1. Given Information

$f\left(x\right)=\frac{2x^2+8x}{x^2-9x+20}andg\left(x\right)=\frac{x-5}{x^2}$

Step 2. Factor each numerator and denominator

$$\begin{gathered} R\left(x\right)=\frac{2x^2+8x}{x^2-9x+20}.\frac{x-5}{x^2} \\ R\left(x\right)=\frac{2x\left(x+4\right)}{\left(x-4\right)\left(x-5\right)}.\frac{x-5}{x.x} \end{gathered}$$

Step 3. Multiply numerators and denominators

$R\left(x\right)=\frac{2x\left(x+4\right)\left(x-5\right)}{x.x\left(x-4\right)\left(x-5\right)}$

Step 4. Remove common factors and simplify

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