Which expression is equivalent to [tex]\left(2 g^3 + 4\right)^2[/tex]?

A. [tex]4 g^6 + 16[/tex]
B. [tex]4 g^6 + 16 g^3 + 16[/tex]
C. [tex]4 g^6 + 8[/tex]
D. [tex]4 g^9 + 16 g^3 + 8[/tex]



Answer :

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]