Sure! Let's solve the given algebraic expression step by step:
We are given the expression:
[tex]\[
\left[a^2 b^3 - 6 x y^2 + \frac{3}{5} a^2 b^3 - 18 x y^2 - 30\right] \times \left[\frac{6}{3} a^2 b^3 + \frac{21}{3} a^2 b^3\right]
\][/tex]
### Simplifying the First Term:
First, let's simplify each term within the square brackets:
1. Combining like terms involving [tex]\(a^2 b^3\)[/tex]:
[tex]\[
a^2 b^3 + \frac{3}{5} a^2 b^3
\][/tex]
Factoring out [tex]\(a^2 b^3\)[/tex]:
[tex]\[
a^2 b^3 \left(1 + \frac{3}{5}\right) = a^2 b^3 \left(\frac{5}{5} + \frac{3}{5}\right) = a^2 b^3 \left(\frac{8}{5}\right)
\][/tex]
[tex]\[
= \frac{8}{5} a^2 b^3
\][/tex]
2. Combining like terms involving [tex]\(xy^2\)[/tex]:
[tex]\[
-6 x y^2 - 18 x y^2
\][/tex]
Factoring out [tex]\(xy^2\)[/tex]:
[tex]\[
-24 x y^2
\][/tex]
So, the first term simplifies to:
[tex]\[
\frac{8}{5} a^2 b^3 - 24 x y^2 - 30
\][/tex]
### Simplifying the Second Term:
Next, let's simplify the second term within the square brackets:
1. Simplifying each term individually:
[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. Combining like terms:
[tex]\[
2 a^2 b^3 + 7 a^2 b^3 = 9 a^2 b^3
\][/tex]
So, the second term simplifies to:
[tex]\[
9 a^2 b^3
\][/tex]
### Combining Both Terms:
Now, multiplying the two simplified terms together:
[tex]\[
\left( \frac{8}{5} a^2 b^3 - 24 x y^2 - 30 \right) \times \left( 9 a^2 b^3 \right)
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\left( \frac{8}{5} a^2 b^3 - 24 x y^2 - 30 \right) \times \left( 9 a^2 b^3 \right)
\][/tex]
