Bot GPT 4-o mini To find the product of \((3x^2 - 8)(4x^2 + 7)\), we will use the distributive property (the FOIL method for binomials).
We will multiply each term in the first polynomial by each term in the second polynomial:
\[ (3x^2)(4x^2) + (3x^2)(7) + (-8)(4x^2) + (-8)(7) \]
Now let's compute each term:
- \(3x^2 \cdot 4x^2 = 12x^4\)
- \(3x^2 \cdot 7 = 21x^2\)
- \(-8 \cdot 4x^2 = -32x^2\)
- \(-8 \cdot 7 = -56\)
Now, we can combine like terms:
- The \(x^2\) terms: \(21x^2 - 32x^2 = -11x^2\)
Now, let's write the full expression, gathering everything together:
\[ (3x^2 - 8)(4x^2 + 7) = 12x^4 - 11x^2 - 56 \]
Putting it in descending order of exponents:
\[ 12x^4 - 11x^2 - 56 \]
Thus, the final answer is:
\[ \boxed{12x^4 - 11x^2 - 56} \]
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