Simplify (6x^4-54x^2)/(2x^2+2x)*(x^2-3x-4)/(x^2-7x+12)

Factor ${\displaystyle 6x^2}$ out of ${\displaystyle 6x^4-54x^2}$.

Rewrite ${\displaystyle 9}$ as ${\displaystyle 3^2}$.

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 ${\displaystyle b=3}$.

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

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

Factor ${\displaystyle 2x}$ out of ${\displaystyle 2x\left(x\right)+2x\left(1\right)}$.

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.

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 ${\displaystyle -7}$.

Write the factored form using these integers.

Combine.

Cancel the common factor of ${\displaystyle 6}$ and ${\displaystyle 2}$.

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

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

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

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

Apply the distributive property.

Move ${\displaystyle x}$.

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