To demonstrate the Closure Property of multiplication of polynomials, we need to multiply the two given polynomials:
\[
\left(\frac{1}{2}x^2 - 3\right) \left(4y^3 + 5x^2\right).
\]
We will use the distributive property (also known as the FOIL method for binomials) to multiply these polynomials:
1. Multiply \(\frac{1}{2}x^2\) by each term in the second polynomial:
\[
\frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3,
\]
\[
\frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4.
\]
2. Multiply \(-3\) by each term in the second polynomial:
\[
-3 \cdot 4y^3 = -12y^3,
\]
\[
-3 \cdot 5x^2 = -15x^2.
\]
Now we can combine all these results:
\[
2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2.
\]
This product is a polynomial since it is expressed as a finite sum of terms with non-negative integer exponents of \(x\) and \(y\).
Among the options given, the correct product that matches what we computed is:
\[
2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2.
\]
So the response demonstrating the Closure Property of multiplication of polynomials is:
**2x²y³ + \(\frac{5}{2}\)x⁴ - 12y³ - 15x².**