To simplify the expression [tex]\(2 a^2 b^3 \left( 4 a^2 + 3 a b^2 - a b \right)\)[/tex], we'll use the distributive property, which states that [tex]\(a(b+c) = ab + ac\)[/tex]. We'll distribute [tex]\(2 a^2 b^3\)[/tex] across each term inside the parentheses.
Let's start by expanding each term inside the parentheses:
[tex]\[ 2 a^2 b^3 (4 a^2) \][/tex]
[tex]\[ 2 a^2 b^3 (3 a b^2) \][/tex]
[tex]\[ 2 a^2 b^3 (-a b) \][/tex]
Now, we'll multiply each term one by one:
1. [tex]\( 2 a^2 b^3 \cdot 4 a^2 \)[/tex]
[tex]\[
= 2 \cdot 4 \cdot a^2 \cdot a^2 \cdot b^3
= 8 a^4 b^3
\][/tex]
2. [tex]\( 2 a^2 b^3 \cdot 3 a b^2 \)[/tex]
[tex]\[
= 2 \cdot 3 \cdot a^2 \cdot a \cdot b^3 \cdot b^2
= 6 a^3 b^5
\][/tex]
3. [tex]\( 2 a^2 b^3 \cdot (-a b) \)[/tex]
[tex]\[
= 2 \cdot (-1) \cdot a^2 \cdot a \cdot b^3 \cdot b
= -2 a^3 b^4
\][/tex]
Now, combine all the expanded terms together:
[tex]\[
8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4
\][/tex]
Now, match this solution with the provided multiple choice options:
A. [tex]\( 8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4 \)[/tex]
B. [tex]\( 8 a^4 b^5 + 3 a^3 b^5 + 2 a^3 b^4 \)[/tex]
C. [tex]\( 8 a^4 b^5 + 3 a^3 b^5 - 2 a^3 b^4 \)[/tex]
D. [tex]\( 8 a^4 b^3 + 6 a^3 b^5 + 2 a^3 b^4 \)[/tex]
We see that Option A matches our simplified expression exactly.
Therefore, the answer is:
[tex]\[
\boxed{A}
\][/tex]
