To expand the expression [tex]\((k^2 - 5k - 2)(k^2 + 2)\)[/tex], we can use the distributive property (also known as the FOIL method for binomials). Here’s the step-by-step expansion:
First, we will distribute [tex]\(k^2\)[/tex] from the first polynomial to each term in the second polynomial:
[tex]\[
k^2 \cdot (k^2 + 2) = k^4 + 2k^2
\][/tex]
Next, we will distribute [tex]\(-5k\)[/tex] from the first polynomial to each term in the second polynomial:
[tex]\[
-5k \cdot (k^2 + 2) = -5k^3 - 10k
\][/tex]
Lastly, we will distribute [tex]\(-2\)[/tex] from the first polynomial to each term in the second polynomial:
[tex]\[
-2 \cdot (k^2 + 2) = -2k^2 - 4
\][/tex]
Now, we combine all these results together:
[tex]\[
k^4 + 2k^2 - 5k^3 - 10k - 2k^2 - 4
\][/tex]
Next, we combine like terms:
[tex]\[
\begin{align*}
&= k^4 - 5k^3 + (2k^2 - 2k^2) - 10k - 4 \\
&= k^4 - 5k^3 - 10k - 4.
\end{align*}
\][/tex]
So, the expanded form of [tex]\((k^2 - 5k - 2)(k^2 + 2)\)[/tex] is:
[tex]\[
\boxed{k^4 - 5k^3 - 10k - 4}.
\][/tex]
