Simplify (2x^2+2x-4)/(2x^2-4x+2)

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

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

Factor ${\displaystyle 2}$ out of ${\displaystyle 2\left(x^2\right)+2\left(x\right)}$.

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

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

Cancel the common factors.

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 ${\displaystyle -2}$ and whose sum is ${\displaystyle 1}$.

Write the factored form using these integers.

Rewrite ${\displaystyle 1}$ as ${\displaystyle 1^2}$.

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 ${\displaystyle b=1}$.

Factor ${\displaystyle x-1}$ out of ${\displaystyle {(x-1)}^2}$.

Cancel the common factor.

Rewrite the expression.