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Answer :
To multiply the polynomials [tex]\((4x^2 + 3x + 7)(8x - 5)\)[/tex], we will distribute each term in the first polynomial by each term in the second polynomial.
1. Distribute [tex]\(4x^2\)[/tex]:
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(32x^3\)[/tex].
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-20x^2\)[/tex].
2. Distribute [tex]\(3x\)[/tex]:
- Multiply [tex]\(3x\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(24x^2\)[/tex].
- Multiply [tex]\(3x\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-15x\)[/tex].
3. Distribute [tex]\(7\)[/tex]:
- Multiply [tex]\(7\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(56x\)[/tex].
- Multiply [tex]\(7\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-35\)[/tex].
4. Combine like terms:
- For the [tex]\(x^2\)[/tex] terms: [tex]\(-20x^2 + 24x^2 = 4x^2\)[/tex].
- For the [tex]\(x\)[/tex] terms: [tex]\(-15x + 56x = 41x\)[/tex].
Combine all parts together, and the final polynomial is:
[tex]\[ 32x^3 + 4x^2 + 41x - 35 \][/tex]
Thus, the solution to the problem is:
[tex]\[
\text{B. } 32x^3 + 4x^2 + 41x - 35
\][/tex]
1. Distribute [tex]\(4x^2\)[/tex]:
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(32x^3\)[/tex].
- Multiply [tex]\(4x^2\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-20x^2\)[/tex].
2. Distribute [tex]\(3x\)[/tex]:
- Multiply [tex]\(3x\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(24x^2\)[/tex].
- Multiply [tex]\(3x\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-15x\)[/tex].
3. Distribute [tex]\(7\)[/tex]:
- Multiply [tex]\(7\)[/tex] by [tex]\(8x\)[/tex] to get [tex]\(56x\)[/tex].
- Multiply [tex]\(7\)[/tex] by [tex]\(-5\)[/tex] to get [tex]\(-35\)[/tex].
4. Combine like terms:
- For the [tex]\(x^2\)[/tex] terms: [tex]\(-20x^2 + 24x^2 = 4x^2\)[/tex].
- For the [tex]\(x\)[/tex] terms: [tex]\(-15x + 56x = 41x\)[/tex].
Combine all parts together, and the final polynomial is:
[tex]\[ 32x^3 + 4x^2 + 41x - 35 \][/tex]
Thus, the solution to the problem is:
[tex]\[
\text{B. } 32x^3 + 4x^2 + 41x - 35
\][/tex]
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