To determine which expression is equivalent to [tex]\(\left(2 g^3 + 4\right)^2\)[/tex], we need to carefully expand this expression step by step.
Given expression:
[tex]\[
\left(2 g^3 + 4\right)^2
\][/tex]
We use the binomial expansion formula, [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex], where:
- [tex]\(a = 2 g^3\)[/tex]
- [tex]\(b = 4\)[/tex]
Now, let's apply the binomial expansion formula:
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
(2 g^3)^2 = 4 g^6
\][/tex]
2. Calculate [tex]\(2ab\)[/tex]:
[tex]\[
2 \cdot (2 g^3) \cdot 4 = 16 g^3
\][/tex]
3. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
4^2 = 16
\][/tex]
Combine these results together:
[tex]\[
\left(2 g^3 + 4\right)^2 = 4 g^6 + 16 g^3 + 16
\][/tex]
Therefore, the expression equivalent to [tex]\(\left(2 g^3 + 4\right)^2\)[/tex] is:
[tex]\[
4 g^6 + 16 g^3 + 16
\][/tex]
From the given options, the correct one is:
[tex]\[
4 g^6 + 16 g^3 + 16
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{2}
\][/tex]
