To determine which expression is equivalent to [tex]\(\left(2 g^3 + 4\right)^2\)[/tex], we need to expand this expression step by step.
Consider the expression [tex]\(\left(2 g^3 + 4\right)^2\)[/tex]. This is a binomial squared, and we can expand it using the binomial theorem or by recognizing it as [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex].
Here:
- [tex]\(a = 2 g^3\)[/tex]
- [tex]\(b = 4\)[/tex]
So, we apply the formula:
[tex]\[ (2 g^3 + 4)^2 = (2 g^3)^2 + 2(2 g^3)(4) + 4^2 \][/tex]
Let's break this down into parts:
1. [tex]\((2 g^3)^2 = (2^2) (g^3)^2 = 4 g^6\)[/tex]
2. [tex]\(2(2 g^3)(4) = 2 \cdot 2 \cdot g^3 \cdot 4 = 16 g^3\)[/tex]
3. [tex]\(4^2 = 16\)[/tex]
Now, we sum these parts:
[tex]\[ 4 g^6 + 16 g^3 + 16 \][/tex]
Therefore, 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]
Comparing this to the given choices, the correct equivalent expression is:
[tex]\[ 4 g^6 + 16 g^3 + 16 \][/tex]
Thus, the expression which is equivalent to [tex]\(\left(2 g^3 + 4\right)^2\)[/tex] is:
[tex]\[ \boxed{4 g^6 + 16 g^3 + 16} \][/tex]
