To find the product of the expressions \(-6a^3b + 2ab^2\) and \(5a^2 - 2ab^2 - b\), we need to distribute each term in the first expression to each term in the second expression and then combine like terms.
Given:
[tex]\[
\left(-6a^3b + 2ab^2\right)\left(5a^2 - 2ab^2 - b\right)
\][/tex]
Let's distribute the terms step by step.
1. Distribute \(-6a^3b\) to each term in \(5a^2 - 2ab^2 - b\):
- \(-6a^3b \cdot 5a^2 = -30a^5b\)
- \(-6a^3b \cdot -2ab^2 = 12a^4b^3\)
- \(-6a^3b \cdot -b = 6a^3b^2\)
2. Distribute \(2ab^2\) to each term in \(5a^2 - 2ab^2 - b\):
- \(2ab^2 \cdot 5a^2 = 10a^3b^2\)
- \(2ab^2 \cdot -2ab^2 = -4a^2b^4\)
- \(2ab^2 \cdot -b = -2ab^3\)
Now, combine all the products:
[tex]\[
-30a^5b + 12a^4b^3 + 6a^3b^2 + 10a^3b^2 - 4a^2b^4 - 2ab^3
\][/tex]
Combine like terms:
- Combine \(6a^3b^2\) and \(10a^3b^2\):
[tex]\[
6a^3b^2 + 10a^3b^2 = 16a^3b^2
\][/tex]
Thus, the final expression is:
[tex]\[
-30a^5b + 12a^4b^3 + 16a^3b^2 - 4a^2b^4 - 2ab^3
\][/tex]
Therefore, the correct answer is:
[tex]\[
-30a^5b + 12a^4b^3 + 16a^3b^2 - 4a^2b^4 - 2ab^3
\][/tex]
So, the correct option is:
[tex]\[
-30 a^5 b+12 a^4 b^3+16 a^3 b^2-4 a^2 b^4-2 a b^3
\][/tex]
