To find the product of [tex]\((3x^2 - 8)(4x^2 + 7)\)[/tex], we'll use the distributive property (also known as the FOIL method for binomials). This involves multiplying each term in the first binomial by each term in the second binomial and then combining like terms. Let us proceed step-by-step:
1. Multiply the terms:
- First terms: [tex]\( 3x^2 \cdot 4x^2 = 12x^4 \)[/tex]
- Outer terms: [tex]\( 3x^2 \cdot 7 = 21x^2 \)[/tex]
- Inner terms: [tex]\( -8 \cdot 4x^2 = -32x^2 \)[/tex]
- Last terms: [tex]\( -8 \cdot 7 = -56 \)[/tex]
2. Combine the like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(21x^2 - 32x^2 = -11x^2\)[/tex]
3. Write the final polynomial:
- [tex]\( 12x^4 \)[/tex]
- [tex]\(-11x^2\)[/tex]
- [tex]\(-56\)[/tex]
Therefore, the product [tex]\((3x^2 - 8)(4x^2 + 7)\)[/tex] is:
[tex]\[
(3x^2 - 8)(4x^2 + 7) = 12x^4 - 11x^2 - 56
\][/tex]
