Simplify (x^2-4)/(x^2+2x-8)*(8x+32)/(x^4+x^3)

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

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 ${\displaystyle b=2}$.

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.

Factor ${\displaystyle 8}$ out of ${\displaystyle 8x+32}$.

Factor ${\displaystyle x^3}$ out of ${\displaystyle x^4+x^3}$.

Combine.

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

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

Move ${\displaystyle 8}$ to the left of ${\displaystyle x+2}$.