Let's break down the given polynomial expression step by step and simplify it:
Given expression:
[tex]\[ (9v^4 + 2) + v^2 (v^2 w^2 + 2v^3 - 2v^2) - (-13v^2 w^3 + 7v^4) \][/tex]
First, let's expand and simplify each part individually.
1. The first part is:
[tex]\[ 9v^4 + 2 \][/tex]
2. The second part involves distributing \( v^2 \):
[tex]\[ v^2(v^2 w^2 + 2v^3 - 2v^2) \][/tex]
Let's distribute \( v^2 \):
[tex]\[ = v^2 \cdot v^2 w^2 + v^2 \cdot 2v^3 - v^2 \cdot 2v^2 \][/tex]
[tex]\[ = v^4 w^2 + 2v^5 - 2v^4 \][/tex]
3. The third part involves distributing the negative sign:
[tex]\[ -(-13v^2 w^3 + 7v^4) \][/tex]
This simplifies to:
[tex]\[ 13v^2 w^3 - 7v^4 \][/tex]
Now let's combine all the parts together:
[tex]\[ (9v^4 + 2) + (v^4 w^2 + 2v^5 - 2v^4) + (13v^2 w^3 - 7v^4) \][/tex]
Next, we combine like terms:
1. Combine \( v^4 \) terms:
[tex]\[ 9v^4 - 2v^4 - 7v^4 = 0 \][/tex]
2. Combine \( v^5 \) terms:
[tex]\[ 2v^5 \][/tex]
3. Combine \( v^4 w^2 \) terms:
[tex]\[ v^4 w^2 \][/tex]
4. Combine \( v^2 w^3 \) terms:
[tex]\[ 13v^2 w^3 \][/tex]
5. The constant term:
[tex]\[ 2 \][/tex]
The combined and simplified expression is:
[tex]\[ 2v^5 + v^4 w^2 + 13v^2 w^3 + 2 \][/tex]
None of the given options have a \( v^5 \) term or match the simplified expression exactly. Thus, it seems there might be an issue with the given options or the provided polynomial expression. However, we should choose the option that has the corrected constants and powers:
Among the provided options:
A. \( 16v^6 w^5 + 2 \)
B. \( v^4 w^2 + 15v^2 w^3 + 2 \)
C. \( 14v^4 + 16v^6 w^5 + 2 \)
D. \( 14v^4 + v^4 w^2 + 15v^2 w^3 + 2 \)
The closest match to our simplified expression, even though it isn't exact (the simplified expression has \( 2 \)),
is:
[tex]\[ \boxed{B. \, v^4 w^2 + 15v^2 w^3 + 2} \][/tex]
