Let's find the simplest form of the expression
[tex]\[
\frac{x+2}{4 x^2+5 x+1} \cdot \frac{4 x+1}{x^2-4}
\][/tex]
First, we need to factorize the denominators and the numerators where possible.
1. Factorize [tex]\(4 x^2 + 5 x + 1\)[/tex]:
[tex]\[
4 x^2 + 5 x + 1 = (4x + 1)(x + 1)
\][/tex]
2. Factorize [tex]\(x^2 - 4\)[/tex]:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
So the expression becomes:
[tex]\[
\frac{x+2}{(4x + 1)(x + 1)} \cdot \frac{4x + 1}{(x - 2)(x + 2)}
\][/tex]
We can cancel out the common factors in the numerator and the denominator. Here, [tex]\(x + 2\)[/tex] in the first numerator and [tex]\(x + 2\)[/tex] in the second denominator cancel out. Similarly, [tex]\(4x + 1\)[/tex] in the second numerator and [tex]\(4x + 1\)[/tex] in the first denominator cancel out. The simplified expression is:
[tex]\[
\frac{1}{(x + 1)(x - 2)}
\][/tex]
Thus, the simplest form of the given expression is:
[tex]\[
\boxed{\frac{1}{(x + 1)(x - 2)}}
\][/tex]
Therefore, the correct answer is:
A. [tex]\(\frac{1}{(x+1)(x-2)}\)[/tex]
