Simplify (x^3+64)/(x^2-16)*(x^2+8x+16)/(x^2-4x+16)

Rewrite ${\displaystyle 64}$ as ${\displaystyle 4^3}$.

Since both terms are perfect cubes, factor using the sum of cubes formula, ${\displaystyle a^3+b^3=(a+b)(a^2-ab+b^2)}$ where ${\displaystyle a=x}$ and ${\displaystyle b=4}$.

Simplify.

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 ${\displaystyle b=4}$.

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

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

Rewrite the polynomial.

Factor using the perfect square trinomial rule ${\displaystyle a^2+2ab+b^2={(a+b)}^2}$, where ${\displaystyle a=x}$ and ${\displaystyle b=4}$.

Cancel the common factor of ${\displaystyle x^2-4x+16}$.

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

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

Raise ${\displaystyle x+4}$ to the power of ${\displaystyle 1}$.

Raise ${\displaystyle x+4}$ to the power of ${\displaystyle 1}$.

Use the power rule ${\displaystyle a^m{a}^n=a^{m+n}}$ to combine exponents.

Add ${\displaystyle 1}$ and ${\displaystyle 1}$.