To expand the expression \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\), we will use the distributive property (also known as the FOIL method for binomials).
Let's distribute each term in the first binomial \(\left(\frac{1}{2}x^2 - 3\right)\) to each term in the second binomial \((4y^3 + 5x^2)\):
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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, we combine all these results:
\[ 2x^2y^3 + \frac{5}{2}x^4 - 12y^3 - 15x^2 \]
Thus, the expanded form of the expression \((\frac{1}{2}x^2 - 3)(4y^3 + 5x^2)\) is:
\[ \frac{5}{2}x^4 + 2x^2y^3 - 15x^2 - 12y^3 \]