To find the product of the given expressions, let's denote them explicitly:
[tex]\[ \text{Expression 1:} \quad \frac{4k + 2}{k^2 - 4} \][/tex]
[tex]\[ \text{Expression 2:} \quad \frac{k - 2}{2k + 1} \][/tex]
We need to multiply these two expressions together.
[tex]\[ \text{Product:} \quad \left( \frac{4k + 2}{k^2 - 4} \right) \cdot \left( \frac{k - 2}{2k + 1} \right) \][/tex]
Before multiplying, it's helpful to factor the denominators and numerators of these expressions, if possible.
Notice that:
[tex]\[ k^2 - 4 = (k - 2)(k + 2) \][/tex]
So we can rewrite the first expression as:
[tex]\[ \frac{4k + 2}{(k - 2)(k + 2)} \][/tex]
The product then becomes:
[tex]\[ \frac{(4k + 2) \cdot (k - 2)}{[(k - 2)(k + 2)] \cdot (2k + 1)} \][/tex]
Now, multiply the numerators and the denominators:
[tex]\[ \text{Numerator:} \quad (4k + 2)(k - 2) \][/tex]
[tex]\[ \text{Denominator:} \quad (k - 2)(k + 2)(2k + 1) \][/tex]
Simplify:
[tex]\[ = \frac{4k(k - 2) + 2(k - 2)}{(k - 2)(k + 2)(2k + 1)} \][/tex]
[tex]\[ = \frac{4k^2 - 8k + 2k - 4}{(k - 2)(k + 2)(2k + 1)} \][/tex]
Combine like terms in the numerator:
[tex]\[ = \frac{4k^2 - 6k - 4}{(k - 2)(k + 2)(2k + 1)} \][/tex]
Factor the numerator where possible. Let's simplify:
[tex]\[ 4k^2 - 6k - 4 = 2(2k^2 - 3k - 2) \][/tex]
Factoring [tex]\(2k^2 - 3k - 2\)[/tex] directly might be complex without additional computation, but given the structure, simplifying directly may show cancellation. Observing factorizations:
[tex]\[ (4k + 2) can be expressed as 2(2k + 1) \][/tex]
Recognize cancellation:
[tex]\[ \frac{2(2k + 1)(k - 2)}{(k - 2)(k + 2)(2k + 1)}\][/tex]
Cancel out common factors [tex]\( (2k + 1) and (k - 2) \)[/tex]:
[tex]\[ = \frac{2}{(k + 2)} \][/tex]
So, the simplified product of the given expressions is:
[tex]\[ \boxed{\frac{2}{k+2}} \][/tex]
Comparing with the multiple options, the correct answer is:
[tex]\[ \frac{2}{k+2} \][/tex]
