Find the Product (x^2+3x-4)/(x^2+4x+4)*(2x^2+4x)/(x^2-4x+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 ${\displaystyle -4}$ and whose sum is ${\displaystyle 3}$.

Write the factored form using these integers.

Rewrite ${\displaystyle 4}$ as ${\displaystyle 2^2}$.

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 ${\displaystyle b=2}$.

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

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

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

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 ${\displaystyle 3}$ and whose sum is ${\displaystyle -4}$.

Write the factored form using these integers.

Combine.

Cancel the common factor of ${\displaystyle x-1}$.

Cancel the common factor of ${\displaystyle x+2}$ and ${\displaystyle {(x+2)}^2}$.

Simplify the expression.