Answer :
To find the product of the two polynomials \( \left(-6 a^3 b + 2 a b^2\right) \) and \( \left(5 a^2 -2 a b^2 - b\right) \), follow these steps:
1. Distribute each term in the first polynomial to each term in the second polynomial.
[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]
[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]
2. Combine the like terms:
[tex]\[ -30 a^5 b \][/tex]
[tex]\[ + 12 a^4 b^3 \][/tex]
[tex]\[ + (6 a^3 b^2 + 10 a^3 b^2) = 16 a^3 b^2 \][/tex]
[tex]\[ -4 a^2 b^4 \][/tex]
[tex]\[ -2 a b^3 \][/tex]
So, the final expanded product of the given polynomials 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 choice 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]
1. Distribute each term in the first polynomial to each term in the second polynomial.
[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]
[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]
2. Combine the like terms:
[tex]\[ -30 a^5 b \][/tex]
[tex]\[ + 12 a^4 b^3 \][/tex]
[tex]\[ + (6 a^3 b^2 + 10 a^3 b^2) = 16 a^3 b^2 \][/tex]
[tex]\[ -4 a^2 b^4 \][/tex]
[tex]\[ -2 a b^3 \][/tex]
So, the final expanded product of the given polynomials 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 choice 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]
