Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials. (1/2 x^2 - 3) (4y^3 + 5x^2)

To demonstrate the Closure Property of multiplication of polynomials, we will multiply the two polynomials \( (1/2 x^2 - 3) \) and \( (4y^3 + 5x^2) \).

Let's compute the product step by step:

1. Distribute \( (1/2 x^2) \) across \( (4y^3 + 5x^2) \):
\[
(1/2 x^2)(4y^3) + (1/2 x^2)(5x^2) = 2x^2y^3 + \frac{5}{2}x^4
\]

2. Distribute \( (-3) \) across \( (4y^3 + 5x^2) \):
\[
-3(4y^3) - 3(5x^2) = -12y^3 - 15x^2
\]

3. Combine all the terms from both distributions:
\[
2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2
\]

Thus, the complete product can be written as:
\[
2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2
\]

Now, let's compare this with the given options:

A) \( \sqrt{x^2y^3 + 4x^4 - 15^2} \)

B) \( 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \)

C) \( \frac{5}{2}x^2y^3 + 2x^4 - 8y^3 + 15x^2 \)

D) \( x^2y^3 + 2x^4 - 8y^3 - 15x^2 \)

The correct option is:

**B) \( 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \)**.