To simplify the given expression:
[tex]\[
\frac{x+2}{4 x^2+5 x+1} \cdot \frac{4 x+1}{x^2-4}
\][/tex]
we should follow these steps:
1. Factor the Denominators:
First, we factor the denominators if possible.
For [tex]\(4 x^2 + 5 x + 1\)[/tex]:
We try to factor it as [tex]\((ax + b)(cx + d)\)[/tex]. We recognize it can be factored as:
[tex]\[
4 x^2 + 5 x + 1 = (4 x + 1)(x + 1)
\][/tex]
For [tex]\(x^2 - 4\)[/tex]:
Recognize it as a difference of squares:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
2. Rewrite the Expression:
Replace the factored forms in the original expression:
[tex]\[
\frac{x+2}{(4x+1)(x+1)} \cdot \frac{4x+1}{(x-2)(x+2)}
\][/tex]
3. Simplify by Canceling Common Factors:
We notice that [tex]\((x+2)\)[/tex] and [tex]\((4x+1)\)[/tex] appear in both numerator and denominator, so they cancel out:
[tex]\[
\frac{\cancel{x+2}}{(4x+1)(x+1)} \cdot \frac{\cancel{4x+1}}{(x-2)\cancel{(x+2)}}
\][/tex]
4. Simplified Form:
After canceling common factors, we get:
[tex]\[
\frac{1}{(x+1)(x-2)}
\][/tex]
5. Match with Given Choices:
The simplified expression [tex]\(\frac{1}{(x+1)(x-2)}\)[/tex] matches option A.
Thus, the simplest form of the given expression is:
[tex]\[
\boxed{\frac{1}{(x+1)(x-2)}}
\][/tex]
