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 will multiply the two given polynomials:

\[
(1/2x^2 - 3) \cdot (4y^3 + 5x^2).
\]

We can use the distributive property (also known as the FOIL method for binomials) to perform the multiplication. We'll distribute each term in the first polynomial by each term in the second polynomial.

1. **Multiply \(1/2x^2\) by \(4y^3\):**
\[
\frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3.
\]

2. **Multiply \(1/2x^2\) by \(5x^2\):**
\[
\frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4.
\]

3. **Multiply \(-3\) by \(4y^3\):**
\[
-3 \cdot 4y^3 = -12y^3.
\]

4. **Multiply \(-3\) by \(5x^2\):**
\[
-3 \cdot 5x^2 = -15x^2.
\]

Now, we will combine all these results:

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

Now, we can write the final product in standard form:

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

Thus, the product of the two polynomials is:

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

This result is clearly a polynomial, demonstrating the Closure Property of multiplication of polynomials, which states that the result of multiplying any two polynomials is also a polynomial.