Simplify (x+2)/(4x^2+5x+1)*(4x+1)/(x^2-4)

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 ${\displaystyle a\cdot c=4\cdot 1=4}$ and whose sum is ${\displaystyle b=5}$.

Factor out the greatest common factor from each group.

Factor the polynomial by factoring out the greatest common factor, ${\displaystyle 4x+1}$.

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 ${\displaystyle 4x+1}$.

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