To find the product that demonstrates the Closure Property of multiplication of polynomials, we will multiply the given polynomials \( (12x^2 - 3)(4y^3 + 5x^2) \).
Let's perform the multiplication using the distributive property (also known as the FOIL method for binomials):
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Distribute \( 12x^2 \) to both terms in the second polynomial:
- \( 12x^2 \cdot 4y^3 = 48x^2y^3 \)
- \( 12x^2 \cdot 5x^2 = 60x^4 \)
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Distribute \(-3\) to both terms in the second polynomial:
- \(-3 \cdot 4y^3 = -12y^3\)
- \(-3 \cdot 5x^2 = -15x^2\)
Now, combining all these terms together, we get:
\[ 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]
Rearranging the terms, we have:
\[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]
Thus, the final product is:
\[ 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \]
However, it seems like none of the provided response options matches this exact result directly. The closest option for the closure property demonstration would include terms up to degree 4 in the polynomial.
If we consider the standard forms, we can identify that the solution involves completing the multiplication and confirming that the resulting polynomial consists of terms that are also polynomials, hence demonstrating the Closure Property. You can look for the polynomial that consists of combinations of \( x^4, x^2y^3, y^3 \) and \( x^2 \) in the options provided.
If any of these response options align with the structure similar to \( 60x^4 + 48x^2y^3 - 12y^3 - 15x^2 \), that would be the correct representation of the Closure Property for given polynomials. Please confirm the options again if possible.