Let's solve the multiplication of [tex]\(\left(3 a^3 - 6 b^3\right)\left(4 a^2 + 5 b^2\right)\)[/tex] step-by-step.
We will use the distributive property (also known as the FOIL method for binomials):
1. Multiply [tex]\(3 a^3\)[/tex] by each term in [tex]\(4 a^2 + 5 b^2\)[/tex]:
- [tex]\(3 a^3 \times 4 a^2 = 12 a^5\)[/tex]
- [tex]\(3 a^3 \times 5 b^2 = 15 a^3 b^2\)[/tex]
2. Multiply [tex]\(-6 b^3\)[/tex] by each term in [tex]\(4 a^2 + 5 b^2\)[/tex]:
- [tex]\(-6 b^3 \times 4 a^2 = -24 a^2 b^3\)[/tex]
- [tex]\(-6 b^3 \times 5 b^2 = -30 b^5\)[/tex]
Now, combine all these products together:
[tex]\[
12 a^5 + 15 a^3 b^2 - 24 a^2 b^3 - 30 b^5
\][/tex]
So, the product of [tex]\(\left(3 a^3 - 6 b^3\right)\left(4 a^2 + 5 b^2\right)\)[/tex] is:
[tex]\[
12 a^5 + 15 a^3 b^2 - 24 a^2 b^3 - 30 b^5
\][/tex]
