Simplify -6/(x^2-16)*(x+2)/(x^2-2x-8)*(16-x^2)/(9x)

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 ${\displaystyle a=4}$ and ${\displaystyle b=x}$.

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.

Move the negative in front of the fraction.

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

Multiply ${\displaystyle \frac{1}{x-4}}$ by ${\displaystyle \frac{6}{(x+4)(x-4)}}$.

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 ${\displaystyle 1}$.

Cancel the common factor of ${\displaystyle 3}$.

Multiply ${\displaystyle \frac{-2}{{(x-4)}^2(x+4)}}$ by ${\displaystyle \frac{(4+x)(4-x)}{3x}}$.

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

Cancel the common factor of ${\displaystyle 4-x}$ and ${\displaystyle {(x-4)}^2}$.

Move ${\displaystyle 3}$ to the left of ${\displaystyle x-4}$.

Reorder factors in ${\displaystyle \frac{2}{3(x-4)x}}$.