Simplify ((-2d^2+d+15)/(9-d^2))÷((4d+1)/(2d^2+11d+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 ${\displaystyle a\cdot c=-2\cdot 15=-30}$ 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, ${\displaystyle -2d-5}$.

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 ${\displaystyle a=3}$ and ${\displaystyle b=d}$.

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

Simplify the expression.

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 ${\displaystyle b=11}$.

Factor out the greatest common factor from each group.

Factor the polynomial by factoring out the greatest common factor, ${\displaystyle 2d+5}$.

Multiply ${\displaystyle \frac{(2d+5)(d+3)}{4d+1}}$ by ${\displaystyle \frac{-2d-5}{3+d}}$.

Factor ${\displaystyle -1}$ out of ${\displaystyle 2d}$.

Rewrite ${\displaystyle 5}$ as ${\displaystyle -1(-5)}$.

Factor ${\displaystyle -1}$ out of ${\displaystyle -(-2d)-1(-5)}$.

Rewrite ${\displaystyle -(-2d-5)}$ as ${\displaystyle -1(-2d-5)}$.

Raise ${\displaystyle -2d-5}$ to the power of ${\displaystyle 1}$.

Raise ${\displaystyle -2d-5}$ to the power of ${\displaystyle 1}$.

Use the power rule ${\displaystyle a^m{a}^n=a^{m+n}}$ to combine exponents.

Add ${\displaystyle 1}$ and ${\displaystyle 1}$.

Reorder terms.

Cancel the common factor.

Rewrite the expression.

Multiply ${\displaystyle -1}$ by ${\displaystyle -1}$.

Multiply ${\displaystyle \frac{{(-2d-5)}^2}{4d+1}}$ by ${\displaystyle 1}$.