Simplify ((5x^2+9x-2)/(2x+6))÷((x^2-4)/(x^2+x-6))

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 ${\displaystyle a\cdot c=5\cdot -2=-10}$ 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, ${\displaystyle 5x-1}$.

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

Factor ${\displaystyle 2}$ out of ${\displaystyle 6}$.

Factor ${\displaystyle 2}$ out of ${\displaystyle 2x+2\cdot 3}$.

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 ${\displaystyle -6}$ 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 ${\displaystyle b=2}$.

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

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

Multiply ${\displaystyle \frac{5x-1}{2}}$ by ${\displaystyle \frac{x-2}{x-2}}$.

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