Simplify (3x^2-27)/(x^2+6x+9)-(2x-4)/(x^2+x-6)

Factor ${\displaystyle 3}$ out of ${\displaystyle 3x^2-27}$.

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 ${\displaystyle b=3}$.

Rewrite ${\displaystyle 9}$ as ${\displaystyle 3^2}$.

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

Rewrite the polynomial.

Factor using the perfect square trinomial rule ${\displaystyle a^2+2ab+b^2={(a+b)}^2}$, where ${\displaystyle a=x}$ and ${\displaystyle b=3}$.

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

Cancel the common factors.

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

Factor ${\displaystyle 2}$ out of ${\displaystyle -4}$.

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

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.

Cancel the common factor.

Rewrite the expression.