Simplify ((2x^3-12x^2)/(x^2-4x-12))÷((8x^3+24x^2)/(x^2+9x+18))

Factor ${\displaystyle 2x^2}$ out of ${\displaystyle 2x^3}$.

Factor ${\displaystyle 2x^2}$ out of ${\displaystyle -12x^2}$.

Factor ${\displaystyle 2x^2}$ out of ${\displaystyle 2x^2\left(x\right)+2x^2(-6)}$.

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 ${\displaystyle -12}$ and whose sum is ${\displaystyle -4}$.

Write the factored form using these integers.

Cancel the common factor.

Rewrite the expression.

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 ${\displaystyle 18}$ and whose sum is ${\displaystyle 9}$.

Write the factored form using these integers.

Factor ${\displaystyle 8x^2}$ out of ${\displaystyle 8x^3+24x^2}$.

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

Multiply ${\displaystyle \frac{1}{x+2}}$ by ${\displaystyle \frac{(x+3)(x+6)}{4(x+3)}}$.

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

Move ${\displaystyle 4}$ to the left of ${\displaystyle x+2}$.