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

The Closure Property of multiplication for polynomials states that the product of two polynomials is also a polynomial.

Let's multiply the two given polynomials:

\[
(1/2 x^2 - 3)(4y^3 + 5x^2)
\]

We can apply the distributive property (also known as the FOIL method for binomials) to perform the multiplication. We distribute each term in the first polynomial to each term in the second polynomial.

1. **Multiply \( \frac{1}{2}x^2 \) with \( 4y^3 \)**:
\[
\frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3
\]

2. **Multiply \( \frac{1}{2}x^2 \) with \( 5x^2 \)**:
\[
\frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4
\]

3. **Multiply \( -3 \) with \( 4y^3 \)**:
\[
-3 \cdot 4y^3 = -12y^3
\]

4. **Multiply \( -3 \) with \( 5x^2 \)**:
\[
-3 \cdot 5x^2 = -15x^2
\]

Now, we combine all these results together:
\[
2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2
\]

We can rearrange the terms in standard polynomial order:
\[
\frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3
\]

The result we obtained is:
\[
\frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3
\]

This expression is indeed a polynomial, which demonstrates the Closure Property of multiplication of polynomials.