To simplify the expression [tex]\(2 a^2 b^3 \left( 4 a^2 + 3 a b^2 - a b \right)\)[/tex], we need to perform the following steps:
1. Distribute the [tex]\(2 a^2 b^3\)[/tex] to each term inside the parentheses:
[tex]\[
2 a^2 b^3 (4 a^2) + 2 a^2 b^3 (3 a b^2) - 2 a^2 b^3 (a b)
\][/tex]
2. Multiply each term individually:
- For the first term [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]
- For the second term [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]
- For the third term [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]
3. Combine all of these results:
[tex]\[
8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4
\][/tex]
Now, comparing this simplified expression to the given 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]\(8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4\)[/tex]
D. [tex]\(8 a^4 b^5 + 3 a^3 b^5 + 2 a^3 b^4\)[/tex]
The correct option that matches the simplified expression is:
C. [tex]\(8 a^4 b^3 + 6 a^3 b^5 - 2 a^3 b^4\)[/tex]
Thus, the best answer for the question is option C.
