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

To demonstrate the Closure Property of multiplication of polynomials, we need to multiply the two given polynomials: \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\).

**Step 1: Use the distributive property (FOIL method) to multiply the polynomials.**

We will multiply each term in the first polynomial by each term in the second polynomial.

1. \(\frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3\)
2. \(\frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4\)
3. \(-3 \cdot 4y^3 = -12y^3\)
4. \(-3 \cdot 5x^2 = -15x^2\)

**Step 2: Combine all the results.**

Now, we gather all the products together:

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

**Step 3: Rearrange the terms by degree.**

For clarity, we can rearrange the terms in standard polynomial form:

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

**Conclusion:**

The result of the multiplication \( \frac{1}{2}x^2 - 3 \) and \( 4y^3 + 5x^2 \) is:

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

Since the result is also a polynomial, we can conclude that the multiplication of these two polynomials demonstrates the Closure Property of multiplication of polynomials.