Simplify (4x^2+23x+15)/(16x^2-9)*(4x^2+9x-9)/(x^2+2x-15)

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 ${\displaystyle a\cdot c=4\cdot 15=60}$ and whose sum is ${\displaystyle b=23}$.

Factor out the greatest common factor from each group.

Factor the polynomial by factoring out the greatest common factor, ${\displaystyle 4x+3}$.

Rewrite ${\displaystyle 16x^2}$ as ${\displaystyle {\left(4x\right)}^2}$.

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 ${\displaystyle a=4x}$ and ${\displaystyle b=3}$.

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 ${\displaystyle a\cdot c=4\cdot -9=-36}$ and whose sum is ${\displaystyle b=9}$.

Factor out the greatest common factor from each group.

Factor the polynomial by factoring out the greatest common factor, ${\displaystyle 4x-3}$.

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 ${\displaystyle -15}$ and whose sum is ${\displaystyle 2}$.

Write the factored form using these integers.

Cancel the common factor of ${\displaystyle x+5}$.

Cancel the common factor of ${\displaystyle 4x-3}$.

Multiply ${\displaystyle \frac{4x+3}{4x+3}}$ by ${\displaystyle \frac{x+3}{x-3}}$.

Cancel the common factor of ${\displaystyle 4x+3}$.