To determine which expression is equivalent to [tex]\(\left(2 g^3 + 4\right)^2\)[/tex], let's follow through with each step of expanding and simplifying it.
1. Identify the given expression:
[tex]\[
(2 g^3 + 4)^2
\][/tex]
2. Apply the binomial theorem for squaring a binomial:
The binomial theorem tells us that [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex]. Here, [tex]\(a = 2g^3\)[/tex] and [tex]\(b = 4\)[/tex].
3. Square each term individually:
[tex]\[
(2g^3)^2 = (2^2) \cdot (g^3)^2 = 4g^6
\][/tex]
[tex]\[
4^2 = 16
\][/tex]
4. Calculate the middle term [tex]\(2ab\)[/tex]:
[tex]\[
2 \cdot (2g^3) \cdot 4 = 2 \cdot 2 \cdot g^3 \cdot 4 = 16g^3
\][/tex]
5. Combine all the terms together:
[tex]\[
(2 g^3 + 4)^2 = (2g^3)^2 + 2 \cdot (2g^3) \cdot 4 + 4^2 = 4g^6 + 16g^3 + 16
\][/tex]
So, the expanded form of the expression [tex]\(\left(2 g^3 + 4\right)^2\)[/tex] is [tex]\(4 g^6 + 16 g^3 + 16\)[/tex].
Thus, the equivalent expression among the options given is:
[tex]\[ 4 g^6 + 16 g^3 + 16 \][/tex]
So, the correct choice is:
[tex]\[ 4 g^6 + 16 g^3 + 16 \][/tex]
