Simplify (4x^3+56x^2+160x)/(x+10)*(81-x^2)/(-x-9)*(3x)/(x^2-5x-36)

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 ${\displaystyle -36}$ and whose sum is ${\displaystyle -5}$.

Write the factored form using these integers.

Factor ${\displaystyle 4x}$ out of ${\displaystyle 4x^3+56x^2+160x}$.

Factor ${\displaystyle x^2+14x+40}$ using the AC method.

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 ${\displaystyle a=9}$ and ${\displaystyle b=x}$.

Cancel the common factor of ${\displaystyle x+10}$.

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

Combine ${\displaystyle 4}$ and ${\displaystyle \frac{(9+x)(9-x)}{-x-9}}$.

Combine ${\displaystyle x}$ and ${\displaystyle \frac{4\left((9+x)(9-x)\right)}{-x-9}}$.

Multiply ${\displaystyle \frac{x\left(4\left((9+x)(9-x)\right)\right)}{-x-9}}$ by ${\displaystyle \frac{3x}{x-9}}$.

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

Multiply ${\displaystyle 3}$ by ${\displaystyle 4}$.

Rewrite ${\displaystyle 9}$ as ${\displaystyle -1(-9)}$.

Factor ${\displaystyle -1}$ out of ${\displaystyle x}$.

Factor ${\displaystyle -1}$ out of ${\displaystyle -1(-9)-1(-x)}$.

Reorder terms.

Cancel the common factor.

Rewrite the expression.

Rewrite ${\displaystyle 9}$ as ${\displaystyle -1(-9)}$.

Factor ${\displaystyle -1}$ out of ${\displaystyle -x}$.

Factor ${\displaystyle -1}$ out of ${\displaystyle -1(-9)-\left(x\right)}$.

Reorder terms.

Cancel the common factor.

Divide ${\displaystyle x^2\left(12(-1\cdot (-1))\right)}$ by ${\displaystyle 1}$.

Multiply ${\displaystyle -1}$ by ${\displaystyle -1}$.

Multiply ${\displaystyle 12}$ by ${\displaystyle 1}$.

Move ${\displaystyle 12}$ to the left of ${\displaystyle x^2}$.