Let's solve the given expression step-by-step.
We need to evaluate the expression:
[tex]\[
\left(a^2 b^3 - 6xy^2 + \frac{3}{5} a^2 b^3 - 18xy^2 - 30\right) \times \left(\frac{6}{3} a^2 b^3 + \frac{21}{3} a^2 b^3\right)
\][/tex]
Take the first term [tex]\(\left(a^2 b^3 - 6xy^2 + \frac{3}{5} a^2 b^3 - 18xy^2 - 30\right)\)[/tex] and simplify it:
1. Combine like terms involving [tex]\(a^2b^3\)[/tex]:
[tex]\[
a^2 b^3 + \frac{3}{5} a^2 b^3 = \left(1 + \frac{3}{5}\right) a^2 b^3 = \frac{8}{5} a^2 b^3
\][/tex]
2. Combine like terms involving [tex]\(xy^2\)[/tex]:
[tex]\[
-6xy^2 - 18xy^2 = -24xy^2
\][/tex]
So, the first term simplifies to:
[tex]\[
\frac{8}{5} a^2 b^3 - 24xy^2 - 30
\][/tex]
Now, take the second term [tex]\(\left(\frac{6}{3} a^2 b^3 + \frac{21}{3} a^2 b^3\right)\)[/tex] and simplify it:
1. Simplify the terms:
[tex]\[
\frac{6}{3} a^2 b^3 = 2 a^2 b^3
\][/tex]
[tex]\[
\frac{21}{3} a^2 b^3 = 7 a^2 b^3
\][/tex]
2. Combine the terms:
[tex]\[
2 a^2 b^3 + 7 a^2 b^3 = 9 a^2 b^3
\][/tex]
Now we need to multiply the simplified first term by the simplified second term:
[tex]\[
\left(\frac{8}{5} a^2 b^3 - 24xy^2 - 30\right) \times \left(9 a^2 b^3\right)
\][/tex]
Distribute [tex]\(9 a^2 b^3\)[/tex] to each term within the parenthesis:
1. Multiply [tex]\(\frac{8}{5} a^2 b^3\)[/tex] by [tex]\(9 a^2 b^3\)[/tex]:
[tex]\[
\frac{8}{5} a^2 b^3 \times 9 a^2 b^3 = \frac{8 \times 9}{5} a^{4} b^{6} = \frac{72}{5} a^4 b^6
\][/tex]
2. Multiply [tex]\(-24xy^2\)[/tex] by [tex]\(9 a^2 b^3\)[/tex]:
[tex]\[
-24xy^2 \times 9 a^2 b^3 = -216 a^2 b^3 xy^2
\][/tex]
3. Multiply [tex]\(-30\)[/tex] by [tex]\(9 a^2 b^3\)[/tex]:
[tex]\[
-30 \times 9 a^2 b^3 = -270 a^2 b^3
\][/tex]
So, the final expanded form of the expression is:
[tex]\[
\frac{72}{5} a^4 b^6 - 216 a^2 b^3 xy^2 - 270 a^2 b^3
\][/tex]
