To find the product of the two polynomials \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\), we will use the distributive property (also known as the FOIL method for binomials).
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Distribute \(\frac{1}{2}x^2\) to both terms in the second polynomial: \[ \frac{1}{2}x^2 \cdot 4y^3 = 2x^2y^3 \] \[ \frac{1}{2}x^2 \cdot 5x^2 = \frac{5}{2}x^4 \]
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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, we combine all these results: \[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]
Thus, the final expression is: \[ \frac{5}{2}x^4 + 2x^2y^3 - 12y^3 - 15x^2 \]
This matches the answer selection of: \(2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2\)
So, the product that demonstrates the Closure Property of multiplication of polynomials is: \(2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2\).