To solve the expression [tex]\( 15 w^2 - \frac{4}{5}\left(5 w^2 - 15\right) + 4 w \)[/tex] for its equivalent form, follow these steps:
1. Expand the expression inside the parentheses:
[tex]\[
\frac{4}{5}\left(5 w^2 - 15\right)
\][/tex]
Multiply both terms inside the parentheses by [tex]\(\frac{4}{5}\)[/tex]:
[tex]\[
\frac{4}{5} \cdot 5 w^2 - \frac{4}{5} \cdot 15
\][/tex]
This simplifies to:
[tex]\[
4 w^2 - 12
\][/tex]
2. Substitute this result back into the original expression:
[tex]\[
15 w^2 - (4 w^2 - 12) + 4 w
\][/tex]
Expand to remove the parentheses:
[tex]\[
15 w^2 - 4 w^2 + 12 + 4 w
\][/tex]
3. Combine like terms:
Group the [tex]\(w^2\)[/tex] terms and the constant terms:
[tex]\[
(15 w^2 - 4 w^2) + 4 w + 12
\][/tex]
Simplify:
[tex]\[
11 w^2 + 4 w + 12
\][/tex]
Therefore, the expression [tex]\(15 w^2 - \frac{4}{5}\left(5 w^2 - 15\right) + 4 w\)[/tex] is equivalent to [tex]\(\boxed{11 w^2 + 4 w + 12}\)[/tex].
So, the correct answer is:
B. [tex]\( 11 w^2 + 4 w + 12 \)[/tex]
