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 will follow these steps:
1. Factor the denominators and numerators where possible:
- The first denominator [tex]\(4 x^2 + 5 x + 1\)[/tex] can be factored using factoring techniques.
[tex]\[
4 x^2 + 5 x + 1 = (4 x + 1)(x + 1)
\][/tex]
- The second denominator [tex]\(x^2 - 4\)[/tex] is a difference of squares and can be factored as:
[tex]\[
x^2 - 4 = (x + 2)(x - 2)
\][/tex]
2. Rewrite the expression with factored forms:
Substituting these factored forms into the expression, we get:
[tex]\[
\frac{x+2}{(4 x + 1)(x + 1)} \cdot \frac{4 x + 1}{(x + 2)(x - 2)}
\][/tex]
3. Simplify the expression by canceling common factors:
- Notice that [tex]\((x + 2)\)[/tex] appears in both the numerator and denominator, so we can cancel [tex]\((x + 2)\)[/tex]:
[tex]\[
\frac{1}{(4 x + 1)(x + 1)} \cdot \frac{4 x + 1}{(x - 2)}
\][/tex]
- Now, the [tex]\((4 x + 1)\)[/tex] term also appears in both the numerator and denominator, so we can cancel [tex]\((4 x + 1)\)[/tex]:
[tex]\[
\frac{1}{(x + 1)} \cdot \frac{1}{(x - 2)} = \frac{1}{(x + 1)(x - 2)}
\][/tex]
Therefore, the simplified expression is
[tex]\[
\frac{1}{(x + 1)(x - 2)}.
\][/tex]
The correct answer is
[tex]\[
\boxed{\frac{1}{(x+1)(x-2)}}.
\][/tex]
