Simplify ((x^2+x-12)/(x-2))/((3x^2+11x-4)/(x^2-4))

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 ${\displaystyle 1}$.

Write the factored form using these integers.

Rewrite ${\displaystyle 4}$ as ${\displaystyle 2^2}$.

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 ${\displaystyle b=2}$.

For a polynomial of the form ${\displaystyle a{x}^2+bx+c}$, rewrite the middle term as a sum of two terms whose product is ${\displaystyle a\cdot c=3\cdot -4=-12}$ and whose sum is ${\displaystyle b=11}$.

Factor out the greatest common factor from each group.

Factor the polynomial by factoring out the greatest common factor, ${\displaystyle 3x-1}$.

Factor ${\displaystyle x+4}$ out of ${\displaystyle (x-3)(x+4)}$.

Factor ${\displaystyle x+4}$ out of ${\displaystyle (3x-1)(x+4)}$.

Cancel the common factor.

Rewrite the expression.

Factor ${\displaystyle x-2}$ out of ${\displaystyle (x+2)(x-2)}$.

Cancel the common factor.

Rewrite the expression.