To find the product of the polynomials \( (3x^2 - 8)(4x^2 + 7) \), we will use the distributive property (also known as the FOIL method for binomials).
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Multiply \( 3x^2 \) by each term in \( (4x^2 + 7) \): \[ 3x^2 \cdot 4x^2 = 12x^4 \] \[ 3x^2 \cdot 7 = 21x^2 \]
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Multiply \( -8 \) by each term in \( (4x^2 + 7) \): \[ -8 \cdot 4x^2 = -32x^2 \] \[ -8 \cdot 7 = -56 \]
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Now, combine all the terms: \[ 12x^4 + 21x^2 - 32x^2 - 56 \]
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Combine like terms (the \( x^2 \) terms): \[ 21x^2 - 32x^2 = -11x^2 \]
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Put it all together in descending order of exponents: \[ 12x^4 - 11x^2 - 56 \]
Thus, the product of \( (3x^2 - 8)(4x^2 + 7) \) is: \[ \boxed{12x^4 - 11x^2 - 56} \]