To simplify the expression \((3x^3 - 2)(x^2 + 4)\), we will use the distributive property (also known as the FOIL method for binomials).
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Distribute \(3x^3\) to both terms in the second polynomial: \[ 3x^3 \cdot x^2 = 3x^5 \] \[ 3x^3 \cdot 4 = 12x^3 \]
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Distribute \(-2\) to both terms in the second polynomial: \[ -2 \cdot x^2 = -2x^2 \] \[ -2 \cdot 4 = -8 \]
Now, combine all the terms together: \[ 3x^5 + 12x^3 - 2x^2 - 8 \]
Thus, the simplified expression is: \[ 3x^5 + 12x^3 - 2x^2 - 8 \]
Looking at the options provided, we see that this is equivalent to option A.
So the final answer is: A: \(3x^5 + 12x^3 - 2x^2 - 8\)
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