Simplify (9/(x^2-14x+45)*(x-5)/(x+9))÷((4x)/(9x^3-729x))

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 ${\displaystyle -14}$.

Write the factored form using these integers.

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

Multiply ${\displaystyle \frac{9}{x-9}}$ by ${\displaystyle \frac{1}{x+9}}$.

Factor ${\displaystyle 9x}$ out of ${\displaystyle 9x^3-729x}$.

Rewrite ${\displaystyle 81}$ as ${\displaystyle 9^2}$.

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 ${\displaystyle b=9}$.

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

Combine ${\displaystyle 9}$ and ${\displaystyle \frac{9x}{4x}}$.

Multiply ${\displaystyle 9}$ by ${\displaystyle 9}$.

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