Use multiplication to find the product that demonstrates the Closure Property of multiplication of polynomials.left parenthesis Start Fraction 1 over 2 End Fraction x squared minus 3 right parenthesis left parenthesis 4 x cubed plus 5 x squared right parenthesis

To demonstrate the Closure Property of multiplication of polynomials, we will multiply the two given polynomials:

\[
\left( \frac{1}{2} x^2 - 3 \right) \left( 4 x^3 + 5 x^2 \right)
\]

We can use the distributive property (also known as the FOIL method for binomials) to perform this multiplication.

1. Multiply \(\frac{1}{2} x^2\) by each term in the second polynomial:
- \(\frac{1}{2} x^2 \cdot 4 x^3 = 2 x^5\)
- \(\frac{1}{2} x^2 \cdot 5 x^2 = \frac{5}{2} x^4\)

2. Multiply \(-3\) by each term in the second polynomial:
- \(-3 \cdot 4 x^3 = -12 x^3\)
- \(-3 \cdot 5 x^2 = -15 x^2\)

Now, combine all the terms together:

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

The final result of the multiplication is:

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

This resulting expression is a polynomial, demonstrating the Closure Property of multiplication of polynomials—when you multiply two polynomials, the product is also a polynomial.