To find the product of the expressions \( (3x^2 - 8)(4x^2 + 7) \), we can use the distributive property (also known as the FOIL method for binomials).
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Distribute \(3x^2\): \[ 3x^2 \cdot 4x^2 = 12x^4 \] \[ 3x^2 \cdot 7 = 21x^2 \]
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Distribute \(-8\): \[ -8 \cdot 4x^2 = -32x^2 \] \[ -8 \cdot 7 = -56 \]
Now, combine all these results together: \[ (3x^2 - 8)(4x^2 + 7) = 12x^4 + 21x^2 - 32x^2 - 56 \]
Next, combine the like terms: \[ 21x^2 - 32x^2 = -11x^2 \]
Putting it all together, we get: \[ 12x^4 - 11x^2 - 56 \]
Thus, the final result in descending order of exponents is: \[ \boxed{12x^4 - 11x^2 - 56} \]