Question

multiply the rational expressions. write your answer in simplest form. $\frac{x^{2}-9x + 18}{5x + 20}cdot\frac{2x + 8}{x^{2}-9}$
$\frac{(x - 6)}{5}cdot\frac{2}{(x + 3)}=\frac{2(x - 6)}{5(x + 3)}$
$\frac{(x + 6)}{5}cdot\frac{2}{(x - 3)}=\frac{2(x + 6)}{5(x - 3)}$
$\frac{-9(x - 2)}{5}cdot\frac{2}{-9}=\frac{2(x - 2)}{5}$
$\frac{(x - 2)}{5(x + 4)}cdot\frac{2(x + 4)}{(x + 1)}=\frac{2(x - 2)(x + 4)}{5(x + 4)(x + 1)}$
steps: 1. factor all terms 2. reduce common terms 3. multiply & write in simplest factored form

Explanation:

Step1: Factor the expressions

• Factor $x^{2}-9x + 18$: $x^{2}-9x + 18=(x - 6)(x - 3)$
• Factor $2x + 8$: $2x+8 = 2(x + 4)$
• Factor $5x + 20$: $5x+20=5(x + 4)$
• Factor $x^{2}-9$: $x^{2}-9=(x + 3)(x - 3)$

So the original expression $\frac{x^{2}-9x + 18}{5x + 20}\cdot\frac{2x + 8}{x^{2}-9}=\frac{(x - 6)(x - 3)}{5(x + 4)}\cdot\frac{2(x + 4)}{(x + 3)(x - 3)}$

Step2: Cancel out the common terms

Cancel out the common terms $(x - 3)$ and $(x + 4)$ in the numerator and denominator.
We get $\frac{(x - 6)\cdot2}{5\cdot(x + 3)}=\frac{2(x - 6)}{5(x + 3)}$

Answer:

$\frac{2(x - 6)}{5(x + 3)}$