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 polynomials \( (12x^2 - 3) \) and \( (4y^3 + 5x^2) \).

Let's perform the multiplication step-by-step:

1. Distribute \( 12x^2 \):
- \( 12x^2 \cdot 4y^3 = 48x^2y^3 \)
- \( 12x^2 \cdot 5x^2 = 60x^4 \)

2. Distribute \( -3 \):
- \( -3 \cdot 4y^3 = -12y^3 \)
- \( -3 \cdot 5x^2 = -15x^2 \)

Now, adding all these results together:
\[
48x^2y^3 + 60x^4 - 12y^3 - 15x^2
\]

Rearranging the terms, we can write it as:
\[
60x^4 + 48x^2y^3 - 12y^3 - 15x^2
\]

The expression we get is:
\[
60x^4 + 48x^2y^3 - 12y^3 - 15x^2
\]

The closest match to this result from your options is:
- \(52x^2y^3 + 2x^4 - 8y^3 + 15x^2 \) (which does not match)
- \(x^2\sqrt{y^3} + 4 x^4 - 15x^2\) (which does not match)
- \(2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2\) (which does not match)

None of the provided options appear to match the calculated product \( 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \). Since it is not available, we can conclude that the options do not correctly represent the product of the two polynomials.

The Closure Property demonstrates that when you multiply two polynomials, the result (the product) is also a polynomial. In this case, our result is indeed a polynomial, confirming the Closure Property for this multiplication.