Simplify (x^2-x-6)/(2x^2+x-6)*(2x^2+7x-15)/(x^2-9)

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 ${\displaystyle -6}$ and whose sum is ${\displaystyle -1}$.

Write the factored form using these integers.

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 ${\displaystyle a\cdot c=2\cdot -6=-12}$ and whose sum is ${\displaystyle b=1}$.

Factor out the greatest common factor from each group.

Factor the polynomial by factoring out the greatest common factor, ${\displaystyle 2x-3}$.

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 ${\displaystyle a\cdot c=2\cdot -15=-30}$ and whose sum is ${\displaystyle b=7}$.

Factor out the greatest common factor from each group.

Factor the polynomial by factoring out the greatest common factor, ${\displaystyle 2x-3}$.

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 ${\displaystyle b=3}$.

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

Cancel the common factor of ${\displaystyle 2x-3}$.

Multiply ${\displaystyle \frac{x+2}{x+2}}$ by ${\displaystyle \frac{x+5}{x+3}}$.

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