Sure, let's solve this step-by-step.
### Step 1: Simplify Each Polynomial Inside the Brackets
#### Left Bracket: [tex]\([a^2 b^3 - 6 y^2 + \frac{3}{5} a^2 b^3 - 18 x y^2 - 30]\)[/tex]
1. Combine like terms:
- [tex]\(a^2 b^3\)[/tex] and [tex]\(\frac{3}{5} a^2 b^3\)[/tex]
[tex]\[
a^2 b^3 + \frac{3}{5} a^2 b^3 = \left(1 + \frac{3}{5}\right) a^2 b^3 = \frac{5}{5} a^2 b^3 + \frac{3}{5} a^2 b^3 = \frac{8}{5} a^2 b^3
\][/tex]
2. Bring all terms together:
[tex]\[
\frac{8}{5} a^2 b^3 - 6 y^2 - 18 x y^2 - 30
\][/tex]
So, the left bracket simplifies to:
[tex]\[
\left[\frac{8}{5} a^2 b^3 - 6 y^2 - 18 x y^2 - 30\right]
\][/tex]
#### Right Bracket: [tex]\[\left[\frac{6}{3} a^2 b^3 + \frac{21}{3} a^2 b^3\right]\][/tex]
Here, each term is already simplified:
1. [tex]\(\frac{6}{3} a^2 b^3\)[/tex] simplifies to [tex]\(2 a^2 b^3\)[/tex].
2. [tex]\(\frac{21}{3} a^2 b^3\)[/tex] simplifies to [tex]\(7 a^2 b^3\)[/tex].
Combine the terms:
[tex]\[
2 a^2 b^3 + 7 a^2 b^3 = 9 a^2 b^3
\][/tex]
So, the right bracket simplifies to:
[tex]\[
\left[9 a^2 b^3\right]
\][/tex]
### Step 2: Multiply the Simplified Expressions
Now we need to multiply the expressions from the left bracket and the right bracket:
[tex]\[
\left[\frac{8}{5} a^2 b^3 - 6 y^2 - 18 x y^2 - 30\right] \times \left[9 a^2 b^3\right]
\][/tex]
Distribute [tex]\(9 a^2 b^3\)[/tex] to each term in the left bracket:
1. [tex]\[
\left(\frac{8}{5} a^2 b^3\right) \times \left(9 a^2 b^3\right)
\][/tex]
[tex]\[
= \frac{8}{5} \times 9 \times a^2 b^3 \times a^2 b^3
\][/tex]
[tex]\[
= \frac{72}{5} a^4 b^6
\][/tex]
2. [tex]\[
-6 y^2 \times 9 a^2 b^3
\][/tex]
[tex]\[
= -54 a^2 b^3 y^2
\][/tex]
3. [tex]\[
-18 x y^2 \times 9 a^2 b^3
\][/tex]
[tex]\[
= -162 x a^2 b^3 y^2
\][/tex]
4. [tex]\[
-30 \times 9 a^2 b^3
\][/tex]
[tex]\[
= -270 a^2 b^3
\][/tex]
### Step 3: Combine All Results
Thus, multiplying the initial polynomials together, we get:
[tex]\[
= \frac{72}{5} a^4 b^6 - 54 a^2 b^3 y^2 - 162 x a^2 b^3 y^2 - 270 a^2 b^3
\][/tex]
This is the simplified result from multiplying the two given polynomials.
So, your final answer is:
[tex]\[
\boxed{\frac{72}{5} a^4 b^6 - 54 a^2 b^3 y^2 - 162 x a^2 b^3 y^2 - 270 a^2 b^3}
\][/tex]
