Alright, let's find the product of the two given rational expressions and simplify it step-by-step:
1. Write the given expressions:
[tex]\[
\frac{4k+2}{k^2-4} \cdot \frac{k-2}{2k+1}
\][/tex]
2. Factor the denominators to simplify them:
Note that [tex]\(k^2 - 4\)[/tex] is a difference of squares and can be factored as:
[tex]\[
k^2 - 4 = (k-2)(k+2)
\][/tex]
So the expression becomes:
[tex]\[
\frac{4k+2}{(k-2)(k+2)} \cdot \frac{k-2}{2k+1}
\][/tex]
3. Combine the two fractions by multiplying the numerators and denominators:
[tex]\[
\frac{(4k+2)(k-2)}{(k-2)(k+2)(2k+1)}
\][/tex]
4. Cancel out the common factors:
Notice that [tex]\((k-2)\)[/tex] appears both in the numerator and the denominator:
[tex]\[
\frac{4k+2}{(k+2)(2k+1)}
\][/tex]
5. Factor the numerator to simplify further:
Factor [tex]\(4k+2\)[/tex]:
[tex]\[
4k + 2 = 2(2k + 1)
\][/tex]
6. Substitute back into the fraction:
[tex]\[
\frac{2(2k+1)}{(k+2)(2k+1)}
\][/tex]
7. Cancel out the common factors again:
Notice that [tex]\((2k+1)\)[/tex] appears in both the numerator and the denominator:
[tex]\[
\frac{2}{k+2}
\][/tex]
Thus, the simplified form of the product [tex]\(\frac{4 k+2}{k^2-4} \cdot \frac{k-2}{2 k+1}\)[/tex] is [tex]\(\boxed{\frac{2}{k+2}}\)[/tex].
