To find the product of the expressions \((3x^2−8)(4x^2+7)\), we will use the distributive property (also known as the FOIL method for binomials). Here are the steps:
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Distribute \(3x^2\) to both terms in the second polynomial:
- \(3x^2 \cdot 4x^2 = 12x^4\)
- \(3x^2 \cdot 7 = 21x^2\)
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Distribute \(-8\) to both terms in the second polynomial:
- \(-8 \cdot 4x^2 = -32x^2\)
- \(-8 \cdot 7 = -56\)
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Now combine all the terms:
- The \(x^4\) term is \(12x^4\).
- Combine the \(x^2\) terms: \(21x^2 - 32x^2 = -11x^2\).
- The constant term is \(-56\).
Putting it all together, the product is: \[ 12x^4 - 11x^2 - 56 \] Thus, the final answer in descending order of exponents is: \[ \boxed{12x^4 - 11x^2 - 56} \]