To multiply the two polynomials \( (3x^2 - 8)(4x^2 + 7) \), we can use the distributive property (also known as the FOIL method for binomials). Here's how to do it step-by-step:
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Distribute \( 3x^2 \) to both terms in the second polynomial: \[ 3x^2 \cdot 4x^2 + 3x^2 \cdot 7 = 12x^4 + 21x^2 \]
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Distribute \( -8 \) to both terms in the second polynomial: \[ -8 \cdot 4x^2 + (-8) \cdot 7 = -32x^2 - 56 \]
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Now, combine all the terms: \[ 12x^4 + 21x^2 - 32x^2 - 56 \]
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Combine the like terms: \[ 12x^4 + (21x^2 - 32x^2) - 56 = 12x^4 - 11x^2 - 56 \]
So the final result is: \[ \boxed{12x^4 - 11x^2 - 56} \]