Simplify ((x^2-4x-45)/(x^2-x-30)*(x^2-36)/(36-x^2))÷((x-9)/(6x))

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 ${\displaystyle -45}$ and whose sum is ${\displaystyle -4}$.

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 ${\displaystyle -30}$ and whose sum is ${\displaystyle -1}$.

Write the factored form using these integers.

Rewrite ${\displaystyle 36}$ as ${\displaystyle 6^2}$.

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 ${\displaystyle b=6}$.

Rewrite ${\displaystyle 36}$ as ${\displaystyle 6^2}$.

Since both terms are perfect squares, factor using the difference of squares formula, ${\displaystyle a^2-b^2=(a+b)(a-b)}$ where ${\displaystyle a=6}$ and ${\displaystyle b=x}$.

Cancel the common factor of ${\displaystyle x-6}$.

Multiply ${\displaystyle \frac{(x-9)(x+5)}{x+5}}$ by ${\displaystyle \frac{x+6}{(6+x)(6-x)}}$.

Cancel the common factor of ${\displaystyle x-9}$.

Combine ${\displaystyle 6}$ and ${\displaystyle \frac{(x+5)(x+6)}{(x+5)\left((6+x)(6-x)\right)}}$.

Combine ${\displaystyle \frac{6\left((x+5)(x+6)\right)}{(x+5)\left((6+x)(6-x)\right)}}$ and ${\displaystyle x}$.

Cancel the common factor of ${\displaystyle x+5}$.

Cancel the common factor of ${\displaystyle x+6}$ and ${\displaystyle 6+x}$.