To simplify the given expression [tex]\( 2 a^2 b^3 (4 a^2 + 3 a b^2 - a b) \)[/tex], let's break it down step by step.
1. Distribute [tex]\( 2 a^2 b^3 \)[/tex] to each term inside the parentheses:
[tex]\[
2 a^2 b^3 \cdot 4 a^2 + 2 a^2 b^3 \cdot 3 a b^2 - 2 a^2 b^3 \cdot a b
\][/tex]
2. Simplify each term independently:
- First term:
[tex]\[
2 a^2 b^3 \cdot 4 a^2 = 2 \cdot 4 \cdot a^2 \cdot a^2 \cdot b^3 = 8 a^4 b^3
\][/tex]
- Second term:
[tex]\[
2 a^2 b^3 \cdot 3 a b^2 = 2 \cdot 3 \cdot a^2 \cdot a \cdot b^3 \cdot b^2 = 6 a^3 b^5
\][/tex]
- Third term:
[tex]\[
2 a^2 b^3 \cdot a b = 2 \cdot a^2 \cdot a \cdot b^3 \cdot b = 2 a^3 b^4
\][/tex]
3. Combine the simplified terms:
[tex]\[
8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4
\][/tex]
Therefore, the simplified expression is:
[tex]\[
8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4
\][/tex]
Comparing this with the given options, the correct answer is:
- Option A. [tex]\( 8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4 \)[/tex]
