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 can apply the distributive property (also known as the FOIL method when dealing with binomials) to expand the expression. Here's a step-by-step breakdown:
### Step 1: Distribute each term in the first polynomial to each term in the second polynomial
We need to multiply each term in [tex]\(-6 a^3 b + 2 a b^2\)[/tex] by each term in [tex]\(5 a^2 - 2 a b^2 - b\)[/tex].
### Step 2: Multiply each pair of terms.
1. [tex]\(-6 a^3 b \cdot 5 a^2 = -30 a^5 b\)[/tex]
2. [tex]\(-6 a^3 b \cdot -2 a b^2 = 12 a^4 b^3\)[/tex]
3. [tex]\(-6 a^3 b \cdot -b = 6 a^3 b^2\)[/tex]
4. [tex]\(2 a b^2 \cdot 5 a^2 = 10 a^3 b^2\)[/tex]
5. [tex]\(2 a b^2 \cdot -2 a b^2 = -4 a^2 b^4\)[/tex]
6. [tex]\(2 a b^2 \cdot -b = -2 a b^3\)[/tex]
### Step 3: Combine like terms.
Adding all the terms 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]
Combine the terms with the same degree:
[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]
### Conclusion:
The product of [tex]\(\left(-6 a^3 b + 2 a b^2\right)\left(5 a^2 - 2 a b^2 - b\right)\)[/tex] 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]
Thus, the option that matches this 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]
