To determine which term is part of the expansion, we use the binomial theorem. The binomial theorem states that the expansion of [tex]\((x + y)^n\)[/tex] is given by
[tex]\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
\][/tex]
where [tex]\(\binom{n}{k}\)[/tex] are the binomial coefficients.
Here, we are asked to expand [tex]\((x + 2)^4\)[/tex].
1. Identify the variables:
- [tex]\(x = x\)[/tex]
- [tex]\(y = 2\)[/tex]
- [tex]\(n = 4\)[/tex]
2. Expand [tex]\((x + 2)^4\)[/tex] using the binomial theorem:
[tex]\[
(x+2)^4 = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} \cdot 2^k
\][/tex]
3. Calculate each term in the expansion:
- For [tex]\(k=0\)[/tex]:
[tex]\[
\binom{4}{0} x^{4-0} \cdot 2^0 = 1 \cdot x^4 \cdot 1 = x^4
\][/tex]
- For [tex]\(k=1\)[/tex]:
[tex]\[
\binom{4}{1} x^{4-1} \cdot 2^1 = 4 \cdot x^3 \cdot 2 = 8x^3
\][/tex]
- For [tex]\(k=2\)[/tex]:
[tex]\[
\binom{4}{2} x^{4-2} \cdot 2^2 = 6 \cdot x^2 \cdot 4 = 24x^2
\][/tex]
- For [tex]\(k=3\)[/tex]:
[tex]\[
\binom{4}{3} x^{4-3} \cdot 2^3 = 4 \cdot x \cdot 8 = 32x
\][/tex]
- For [tex]\(k=4\)[/tex]:
[tex]\[
\binom{4}{4} x^{4-4} \cdot 2^4 = 1 \cdot 1 \cdot 16 = 16
\][/tex]
Combining these terms, we get the full expansion:
[tex]\[
(x+2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16
\][/tex]
From the expanded form, it is clear that one of the terms is [tex]\(24x^2\)[/tex].
Therefore, the correct answer is:
[tex]\[
24 x^2
\][/tex]
