To find the product \((12x^2 - 3)(4y^3 + 5x^2)\), we will use the distributive property (also known as the FOIL method for binomials, where applicable). We will distribute each term in the first polynomial to each term in the second polynomial.
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Multiply \(12x^2\) by \(4y^3\): \[ 12x^2 \cdot 4y^3 = 48x^2y^3 \]
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Multiply \(12x^2\) by \(5x^2\): \[ 12x^2 \cdot 5x^2 = 60x^4 \]
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Multiply \(-3\) by \(4y^3\): \[ -3 \cdot 4y^3 = -12y^3 \]
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Multiply \(-3\) by \(5x^2\): \[ -3 \cdot 5x^2 = -15x^2 \]
Now, we will combine all the results from the products we calculated: \[ 48x^2y^3 + 60x^4 - 12y^3 - 15x^2 \]
Next, we will arrange the terms in standard form: \[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]
So the product of \((12x^2 - 3)(4y^3 + 5x^2)\) is: \[ 60x^4 + 48x^2y^3 - 15x^2 - 12y^3 \]
Thus, the resulting polynomial demonstrates the Closure Property of multiplication of polynomials, as it results in another polynomial.
If we compare this result with the given responses, none of the choices listed match our computed result exactly. However, it's essential to ensure that the correct simplification is provided and understood. If given options were meant for a different polynomial or a mistake was present in the options, verification could be necessary.
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