Simplify (x^4-81)/(2x^2-12x+18)

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 ${\displaystyle b=3}$.

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

Factor ${\displaystyle 2}$ out of ${\displaystyle -12x}$.

Factor ${\displaystyle 2}$ out of ${\displaystyle 18}$.

Factor ${\displaystyle 2}$ out of ${\displaystyle 2x^2+2(-6x)}$.

Factor ${\displaystyle 2}$ out of ${\displaystyle 2(x^2-6x)+2\cdot 9}$.

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 ${\displaystyle 2{(x-3)}^2}$.

Cancel the common factor.

Rewrite the expression.