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Bot GPT 4-o mini To demonstrate the Closure Property of multiplication of polynomials, we need to find the product of the two given polynomials \((12x^2 - 3)(4y^3 + 5x^2)\).
We will use the distributive property (also known as the FOIL method for two binomials) to multiply them:
1. **Multiply \(12x^2\) by \(4y^3\)**:
\[
12x^2 \cdot 4y^3 = 48x^2y^3
\]
2. **Multiply \(12x^2\) by \(5x^2\)**:
\[
12x^2 \cdot 5x^2 = 60x^4
\]
3. **Multiply \(-3\) by \(4y^3\)**:
\[
-3 \cdot 4y^3 = -12y^3
\]
4. **Multiply \(-3\) by \(5x^2\)**:
\[
-3 \cdot 5x^2 = -15x^2
\]
Now, we combine all the terms:
\[
48x^2y^3 + 60x^4 - 12y^3 - 15x^2
\]
Finally, let's arrange the terms in standard polynomial form (typically by decreasing order of exponents):
\[
60x^4 + 48x^2y^3 - 15x^2 - 12y^3
\]
From your provided options, the expression does not match any of them exactly; however, the correct polynomial after multiplying \((12x^2 - 3)(4y^3 + 5x^2)\) represents the Closure Property of multiplication of polynomials.
If we recheck the initial response options for potential matches of the final expression:
- None of the given options are correct as none of them exactly match the final product we calculated.
Thus,
Result: \(60x^4 + 48x^2y^3 - 15x^2 - 12y^3\), demonstrating the Closure Property of multiplication of polynomials.
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