Simplify (x^2-16)/(x^2-2x-8)*(x+2)/x

Rewrite ${\displaystyle 16}$ as ${\displaystyle 4^2}$.

Since both terms are perfect squares, factor using the difference of squares formula, ${\displaystyle a^2-b^2=(a+b)(a-b)}$ where ${\displaystyle a=x}$ and ${\displaystyle b=4}$.

Consider the form ${\displaystyle x^2+bx+c}$. Find a pair of integers whose product is ${\displaystyle c}$ and whose sum is ${\displaystyle b}$. In this case, whose product is ${\displaystyle -8}$ and whose sum is ${\displaystyle -2}$.

Write the factored form using these integers.

Cancel the common factor of ${\displaystyle x+2}$.

Multiply ${\displaystyle \frac{(x+4)(x-4)}{x-4}}$ by ${\displaystyle \frac{1}{x}}$.

Cancel the common factor of ${\displaystyle x-4}$.