Answer :
Let's simplify the given expression step-by-step:
We start with the expression:
[tex]\[ 2 a^2 b^3 \left( 4 a^2 + 3 a b^2 - a b \right) \][/tex]
1. Distribute [tex]\(2 a^2 b^3\)[/tex] into 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. Multiply each 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]
[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]
[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 terms together:
[tex]\[ 8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4 \][/tex]
So, after simplification, the result is:
[tex]\[ 2 a^3 b^3 (4 a + 3 b^2 - b) \][/tex]
which matches with our steps of simplifying the expression.
Thus, the correct option that matches this simplified form is:
[tex]\[ \boxed{8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4} \][/tex]
This corresponds to option (A) from the provided choices.
We start with the expression:
[tex]\[ 2 a^2 b^3 \left( 4 a^2 + 3 a b^2 - a b \right) \][/tex]
1. Distribute [tex]\(2 a^2 b^3\)[/tex] into 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. Multiply each 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]
[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]
[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 terms together:
[tex]\[ 8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4 \][/tex]
So, after simplification, the result is:
[tex]\[ 2 a^3 b^3 (4 a + 3 b^2 - b) \][/tex]
which matches with our steps of simplifying the expression.
Thus, the correct option that matches this simplified form is:
[tex]\[ \boxed{8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4} \][/tex]
This corresponds to option (A) from the provided choices.
