Let's break down the given expression step by step to find the product:
We start with the algebraic expressions:
[tex]\[
\left(-6 a^3 b + 2 a b^2\right)\left(5 a^2 - 2 a b^2 - b\right)
\][/tex]
Let's distribute each term in the first expression by each term in the second expression, and then combine like terms.
Step 1: Multiply [tex]\( -6 a^3 b \)[/tex] by every term in [tex]\( 5 a^2 - 2 a b^2 - b \)[/tex]:
[tex]\[
-6 a^3 b \cdot 5 a^2 = -30 a^5 b
\][/tex]
[tex]\[
-6 a^3 b \cdot -2 a b^2 = 12 a^4 b^3
\][/tex]
[tex]\[
-6 a^3 b \cdot -b = 6 a^3 b^2
\][/tex]
Step 2: Multiply [tex]\( 2 a b^2 \)[/tex] by every term in [tex]\( 5 a^2 - 2 a b^2 - b \)[/tex]:
[tex]\[
2 a b^2 \cdot 5 a^2 = 10 a^3 b^2
\][/tex]
[tex]\[
2 a b^2 \cdot -2 a b^2 = -4 a^2 b^4
\][/tex]
[tex]\[
2 a b^2 \cdot -b = -2 a b^3
\][/tex]
Step 3: Combine all the terms we obtained from steps 1 and 2:
[tex]\[
-30 a^5 b + 12 a^4 b^3 + 6 a^3 b^2 + 10 a^3 b^2 - 4 a^2 b^4 - 2 a b^3
\][/tex]
Step 4: Combine like terms:
[tex]\[
-30 a^5 b + 12 a^4 b^3 + (6 a^3 b^2 + 10 a^3 b^2) - 4 a^2 b^4 - 2 a b^3
\][/tex]
[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]
Thus, the product of the expressions 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]
Therefore, the correct answer is:
[tex]\[
-30a^5b + 12a^4b^3 + 16a^3b^2 - 4a^2b^4 - 2ab^3
\][/tex]
