Answer :
Let's multiply the polynomials [tex]\((7x^2 + 5x + 7)(4x - 6)\)[/tex] step-by-step:
1. Distribute each term in the first polynomial by every term in the second polynomial:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex] to get [tex]\(28x^3\)[/tex].
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-6\)[/tex] to get [tex]\(-42x^2\)[/tex].
- Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex] to get [tex]\(20x^2\)[/tex].
- Multiply [tex]\(5x\)[/tex] by [tex]\(-6\)[/tex] to get [tex]\(-30x\)[/tex].
- Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex] to get [tex]\(28x\)[/tex].
- Multiply [tex]\(7\)[/tex] by [tex]\(-6\)[/tex] to get [tex]\(-42\)[/tex].
2. Combine the results to form a new polynomial:
[tex]\[
28x^3 + (-42x^2) + 20x^2 + (-30x) + 28x + (-42)
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-30x + 28x = -2x\)[/tex].
4. Final polynomial:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
This matches answer choice B: [tex]\(\boxed{28x^3 - 22x^2 - 2x - 42}\)[/tex].
1. Distribute each term in the first polynomial by every term in the second polynomial:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex] to get [tex]\(28x^3\)[/tex].
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-6\)[/tex] to get [tex]\(-42x^2\)[/tex].
- Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex] to get [tex]\(20x^2\)[/tex].
- Multiply [tex]\(5x\)[/tex] by [tex]\(-6\)[/tex] to get [tex]\(-30x\)[/tex].
- Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex] to get [tex]\(28x\)[/tex].
- Multiply [tex]\(7\)[/tex] by [tex]\(-6\)[/tex] to get [tex]\(-42\)[/tex].
2. Combine the results to form a new polynomial:
[tex]\[
28x^3 + (-42x^2) + 20x^2 + (-30x) + 28x + (-42)
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex].
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-30x + 28x = -2x\)[/tex].
4. Final polynomial:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
This matches answer choice B: [tex]\(\boxed{28x^3 - 22x^2 - 2x - 42}\)[/tex].
