To demonstrate the Closure Property of multiplication of polynomials, we need to multiply the given polynomials:
\[ \left(\frac{1}{2}x^2 - 3\right)(4y^3 + 5x^2) \]
We'll distribute the first polynomial over the second:
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Multiply \(\frac{1}{2}x^2\) by \(4y^3\): \[ \frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3 \]
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Multiply \(\frac{1}{2}x^2\) by \(5x^2\): \[ \frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4 \]
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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, combine all these results: \[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]
Thus, the product of the two polynomials is: \[ 2x^2y^3 - 12y^3 + \frac{5}{2}x^4 - 15x^2 \]
From the options provided, we can select the one that matches our result: \[ \frac{5}{2}x^4 + 2x^2y^3 - 12y^3 - 15x^2 \]
This corresponds to: \[ \frac{5}{2} x^4 + 2 x^2 y^3 - 12 y^3 - 15 x^2 \]
So the correct response is: \(\frac{5}{2}x^4 + 2x^2y^3 - 12y^3 - 15x^2\)