To find the product [tex]\(\left(-6 a^3 b + 2 a b^2\right)\left(5 a^2 - 2 a b^2 - b\right)\)[/tex], we will perform polynomial multiplication and combine like terms. Here’s the step-by-step process:
First, we distribute [tex]\( (-6 a^3 b) \)[/tex] across each term inside the second parenthesis:
[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]
Next, we distribute [tex]\( 2 a b^2 \)[/tex] across each term inside the second parenthesis:
[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]
Now, we combine all the products together:
[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]
We then combine the 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]
So, the 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]
Therefore, 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]
