To demonstrate the Closure Property of multiplication of polynomials, we will multiply the two given polynomials:
\[(\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\]
We will perform the multiplication using the distributive property (also known as the FOIL method for binomials).
1. Distribute \(\frac{1}{2}x^2\) to both terms 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. Distribute \(-3\) to both terms in the second polynomial:
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)
3. Combine all the results:
\[
2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2
\]
So the product of the two polynomials is:
\[
2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2
\]
This demonstrates the Closure Property of multiplication of polynomials, as the result is also a polynomial.
Among the provided options, the correct result is:
**2x^2y^3 + 5/2x^4 - 12y^3 - 15x^2**