To demonstrate the Closure Property of multiplication of polynomials, we will multiply the given polynomials \((12x^2 - 3)\) and \((4y^3 + 5x^2)\).
Let's perform the multiplication:
\[ (12x^2 - 3)(4y^3 + 5x^2) \]
We will use the distributive property (often referred to as the FOIL method for binomials) to expand the expression:
- \(12x^2 \cdot 4y^3 = 48x^2y^3\)
- \(12x^2 \cdot 5x^2 = 60x^4\)
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)
Now, we combine all these results:
\[ 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]
Rearranging this to standard form (typically in descending order of degree for each term):
\[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]
Now that we've computed the product, we can note that the final expression \(60x^4 + 48x^2y^3 - 12y^3 - 15x^2\) is indeed a polynomial, demonstrating the Closure Property of multiplication of polynomials; the product of two polynomials is still a polynomial.
Now, comparing this with the options provided, it seems none of the options matches the resulting polynomial we calculated. Please double-check if there are any typographical errors in the choices or if there are other requirements.