To perform the binomial expansion of [tex]\((x + 2)^4\)[/tex], we will use the Binomial Theorem. The Binomial Theorem states that:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
For our particular case [tex]\((x + 2)^4\)[/tex], let's set [tex]\(a = x\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(n = 4\)[/tex]. Applying the binomial theorem, we get:
[tex]\[
(x + 2)^4 = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} \cdot 2^k
\][/tex]
Let's break this down term by term:
1. For [tex]\(k = 0\)[/tex]:
[tex]\[
\binom{4}{0} x^{4-0} \cdot 2^0 = 1 \cdot x^4 \cdot 1 = x^4
\][/tex]
2. For [tex]\(k = 1\)[/tex]:
[tex]\[
\binom{4}{1} x^{4-1} \cdot 2^1 = 4 \cdot x^3 \cdot 2 = 8x^3
\][/tex]
3. For [tex]\(k = 2\)[/tex]:
[tex]\[
\binom{4}{2} x^{4-2} \cdot 2^2 = 6 \cdot x^2 \cdot 4 = 24x^2
\][/tex]
4. For [tex]\(k = 3\)[/tex]:
[tex]\[
\binom{4}{3} x^{4-3} \cdot 2^3 = 4 \cdot x \cdot 8 = 32x
\][/tex]
5. For [tex]\(k = 4\)[/tex]:
[tex]\[
\binom{4}{4} x^{4-4} \cdot 2^4 = 1 \cdot 1 \cdot 16 = 16
\][/tex]
Adding all these terms together, we get:
[tex]\[
x^4 + 8x^3 + 24x^2 + 32x + 16
\][/tex]
Therefore, the binomial expansion of [tex]\((x + 2)^4\)[/tex] is:
[tex]\[
x^4 + 8x^3 + 24x^2 + 32x + 16
\][/tex]
The correct answer from the given choices is:
[tex]\[
x^4 + 8x^3 + 24x^2 + 32x + 16
\][/tex]
