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

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 can use the distributive property (also known as the FOIL method for binomials) to multiply these two polynomials.

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:
\[
-3 \cdot 4y^3 = -12y^3
\]
\[
-3 \cdot 5x^2 = -15x^2
\]

Now, we combine all the terms:

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

The fully multiplied polynomial is:

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

Now we can look for this expression among the provided answer choices. The correct answer is:

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