Simplify ((7-x)/(7-x)*(x^2-1)/(64-x^2))÷((x^2+3x-4)/(x^2+12x+32))

Cancel the common factor of ${\displaystyle 7-x}$.

Multiply ${\displaystyle \frac{x^2-1}{64-x^2}}$ by ${\displaystyle 1}$.

Rewrite ${\displaystyle 1}$ as ${\displaystyle 1^2}$.

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 ${\displaystyle b=1}$.

Rewrite ${\displaystyle 64}$ as ${\displaystyle 8^2}$.

Since both terms are perfect squares, factor using the difference of squares formula, ${\displaystyle a^2-b^2=(a+b)(a-b)}$ where ${\displaystyle a=8}$ and ${\displaystyle b=x}$.

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 ${\displaystyle 32}$ and whose sum is ${\displaystyle 12}$.

Write the factored form using these integers.

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 ${\displaystyle -4}$ and whose sum is ${\displaystyle 3}$.

Write the factored form using these integers.

Cancel the common factor of ${\displaystyle x-1}$.

Multiply ${\displaystyle \frac{x+1}{(8+x)(8-x)}}$ by ${\displaystyle \frac{(x+4)(x+8)}{x+4}}$.

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

Cancel the common factor of ${\displaystyle x+8}$ and ${\displaystyle 8+x}$.