College

Multiply the polynomials:

[tex]\left(7x^2 + 5x + 7\right)(4x - 6)[/tex]

A. [tex]28x^3 - 22x^2 - 2x + 42[/tex]

B. [tex]28x^3 - 22x^2 - 58x - 42[/tex]

C. [tex]28x^3 - 22x^2 - 2x - 42[/tex]

D. [tex]28x^3 - 62x^2 - 2x - 42[/tex]

Answer :

Let's multiply the polynomials [tex]\((7x^2 + 5x + 7)(4x - 6)\)[/tex] together, step by step.

1. Distribute each term in the first polynomial to every term in the second polynomial:

- Start with [tex]\(7x^2 \times (4x - 6)\)[/tex]:
- [tex]\(7x^2 \times 4x = 28x^3\)[/tex]
- [tex]\(7x^2 \times -6 = -42x^2\)[/tex]

- Next, distribute [tex]\(5x\)[/tex] across [tex]\(4x - 6\)[/tex]:
- [tex]\(5x \times 4x = 20x^2\)[/tex]
- [tex]\(5x \times -6 = -30x\)[/tex]

- Lastly, distribute [tex]\(7\)[/tex] across [tex]\(4x - 6\)[/tex]:
- [tex]\(7 \times 4x = 28x\)[/tex]
- [tex]\(7 \times -6 = -42\)[/tex]

2. Combine all the terms:

The expression becomes:
[tex]\(28x^3 + (-42x^2) + 20x^2 + (-30x) + 28x + (-42)\)[/tex].

3. Combine like terms:

- For [tex]\(x^3\)[/tex], we only have one term: [tex]\(28x^3\)[/tex].
- For [tex]\(x^2\)[/tex] terms: [tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex].
- For [tex]\(x\)[/tex] terms: [tex]\(-30x + 28x = -2x\)[/tex].
- The constant term is [tex]\(-42\)[/tex].

Putting it all together, the result of multiplying the polynomials is:
[tex]\[ 28x^3 - 22x^2 - 2x - 42 \][/tex]

Therefore, the correct answer is C. [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].