College

Multiply the polynomials:

[tex]
(4x^2 + 4x + 6)(7x + 5)
[/tex]

A. [tex]28x^3 + 8x^2 + 22x - 30[/tex]

B. [tex]28x^3 - 40x^2 + 70x + 30[/tex]

C. [tex]28x^3 + 8x^2 + 22x + 30[/tex]

D. [tex]28x^3 + 48x^2 + 62x + 30[/tex]

Answer :

To multiply the polynomials [tex]\((4x^2 + 4x + 6)\)[/tex] and [tex]\((7x + 5)\)[/tex], we will use the distributive property, also known as the FOIL method for binomials, but extended for polynomials with more terms.

Here's a detailed step-by-step solution:

1. Distribute Each Term:
- Multiply each term in the first polynomial by each term in the second polynomial.

2. Multiply [tex]\(4x^2\)[/tex] by each term in [tex]\(7x + 5\)[/tex]:
- [tex]\(4x^2 \times 7x = 28x^3\)[/tex]
- [tex]\(4x^2 \times 5 = 20x^2\)[/tex]

3. Multiply [tex]\(4x\)[/tex] by each term in [tex]\(7x + 5\)[/tex]:
- [tex]\(4x \times 7x = 28x^2\)[/tex]
- [tex]\(4x \times 5 = 20x\)[/tex]

4. Multiply [tex]\(6\)[/tex] by each term in [tex]\(7x + 5\)[/tex]:
- [tex]\(6 \times 7x = 42x\)[/tex]
- [tex]\(6 \times 5 = 30\)[/tex]

5. Combine like terms:
- [tex]\(28x^3\)[/tex] (only one term of cubic degree).
- Combine the quadratic terms: [tex]\(20x^2 + 28x^2 = 48x^2\)[/tex].
- Combine the linear terms: [tex]\(20x + 42x = 62x\)[/tex].
- The constant term is [tex]\(30\)[/tex].

So, the result of [tex]\(\left(4x^2 + 4x + 6\right)(7x + 5)\)[/tex] is [tex]\(28x^3 + 48x^2 + 62x + 30\)[/tex].

Therefore, the correct answer is:

D. [tex]\(28x^3 + 48x^2 + 62x + 30\)[/tex]