To find the product of the two expressions \( (-6 a^3 b + 2 a b^2)(5 a^2 - 2 a b^2 - b) \), we need to multiply each term in the first expression by each term in the second expression. Let's break it down step-by-step:
1. Multiply \(-6 a^3 b\) by each term in \( 5 a^2 - 2 a b^2 - b \):
- \((-6 a^3 b) \cdot (5 a^2) = -30 a^5 b \)
- \((-6 a^3 b) \cdot (-2 a b^2) = 12 a^4 b^3 \)
- \((-6 a^3 b) \cdot (-b) = 6 a^3 b^2 \)
2. Multiply \(2 a b^2\) by each term in \(5 a^2 - 2 a b^2 - b\):
- \((2 a b^2) \cdot (5 a^2) = 10 a^3 b^2\)
- \((2 a b^2) \cdot (-2 a b^2) = -4 a^2 b^4\)
- \((2 a b^2) \cdot (-b) = -2 a b^3\)
3. Add all these terms together to get the final expanded expression:
- \( -30 a^5 b \)
- \( + 12 a^4 b^3 \)
- \( + 6 a^3 b^2 \)
- \( + 10 a^3 b^2 \)
- \( -4 a^2 b^4 \)
- \( -2 a b^3 \)
4. Combine like terms:
- \(6 a^3 b^2 + 10 a^3 b^2 = 16 a^3 b^2 \)
Therefore, the final product 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]
Comparing this with the given possible answers, we find that the correct answer 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]
