Use multiplication to find the product that demonstrates the Closure Property of Multiplication of polynomials.

To demonstrate the Closure Property of Multiplication of polynomials using the given expression \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\), we will multiply these two polynomials together.

Let's break it down step-by-step:

1. **Distribute each term in the first polynomial by each term in the second polynomial**:

\[
\left(\frac{1}{2}x^2 - 3\right)(4y^3 + 5x^2) = \left(\frac{1}{2}x^2\right)(4y^3) + \left(\frac{1}{2}x^2\right)(5x^2) - 3(4y^3) - 3(5x^2)
\]

2. **Perform the multiplications**:

- \(\left(\frac{1}{2}x^2\right)(4y^3) = \frac{4}{2}x^2y^3 = 2x^2y^3\)
- \(\left(\frac{1}{2}x^2\right)(5x^2) = \frac{5}{2}x^{2+2} = \frac{5}{2}x^4\)
- \(-3(4y^3) = -12y^3\)
- \(-3(5x^2) = -15x^2\)

3. **Combine all the terms**:

\[
2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2
\]

4. **Reorganize and combine like terms, if applicable**:

The multiplication results in:
\[
\frac{5}{2}x^4 + (2x^2y^3 - 15x^2 - 12y^3)
\]

While there are no like terms to combine further (as we're dealing with different degrees and variables), we can express the final result concisely as:

\[
\frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3
\]

### Conclusion:
The product of the two polynomials is a polynomial expressed by the equation:

\[
\frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3
\]

This demonstrates the Closure Property of Multiplication of polynomials, as the result is also a polynomial.