Sure, let's go through the detailed steps to simplify the given expression.
We need to simplify [tex]\( 2 a^2 b^3 \left( 4 a^2 + 3 a b^2 - a b \right) \)[/tex].
Step 1: Distribute [tex]\(2 a^2 b^3\)[/tex] to each term inside the parentheses:
[tex]\[
2 a^2 b^3 \cdot 4 a^2
\][/tex]
[tex]\[
2 a^2 b^3 \cdot 3 a b^2
\][/tex]
[tex]\[
2 a^2 b^3 \cdot (- a b)
\][/tex]
Step 2: Simplify each term:
- For the first term, multiply:
[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]
- For the second term, multiply:
[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]
- For the third term, multiply:
[tex]\[
2 a^2 b^3 \cdot (- a b) = 2 \cdot (-1) \cdot a^2 \cdot a \cdot b^3 \cdot b = -2 a^3 b^4
\][/tex]
Step 3: Combine all simplified terms:
[tex]\[
8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4
\][/tex]
Conclusion:
Therefore, the simplified expression is [tex]\( 8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4 \)[/tex].
Looking at the given choices, the correct answer is:
A. [tex]\( 8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4 \)[/tex]
