High School

Multiply the polynomials: [tex]\left(4x^2 + 4x + 6\right)(7x + 5)[/tex]

A. [tex]28x^3 + 8x^2 + 22x + 30[/tex]
B. [tex]28x^3 + 48x^2 + 62x + 30[/tex]
C. [tex]28x^3 - 40x^2 + 70x + 30[/tex]
D. [tex]28x^3 + 8x^2 + 22x - 30[/tex]

Answer :

Sure! Let's multiply the polynomials step-by-step:

We're given the polynomials [tex]\((4x^2 + 4x + 6)\)[/tex] and [tex]\((7x + 5)\)[/tex]. We'll distribute each term in the first polynomial to every term in the second polynomial.

1. Distribute [tex]\(4x^2\)[/tex]:

- [tex]\(4x^2 \times 7x = 28x^3\)[/tex]
- [tex]\(4x^2 \times 5 = 20x^2\)[/tex]

After this step, we have: [tex]\(28x^3 + 20x^2\)[/tex].

2. Distribute [tex]\(4x\)[/tex]:

- [tex]\(4x \times 7x = 28x^2\)[/tex]
- [tex]\(4x \times 5 = 20x\)[/tex]

Adding these terms to the previous result:
[tex]\[28x^3 + 20x^2 + 28x^2 + 20x\][/tex]

3. Distribute [tex]\(6\)[/tex]:

- [tex]\(6 \times 7x = 42x\)[/tex]
- [tex]\(6 \times 5 = 30\)[/tex]

Adding these terms as well:
[tex]\[28x^3 + 20x^2 + 28x^2 + 20x + 42x + 30\][/tex]

4. Combine like terms:

- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(20x^2 + 28x^2 = 48x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(20x + 42x = 62x\)[/tex]

Finally, our expression is:
[tex]\[28x^3 + 48x^2 + 62x + 30\][/tex]

So the correct answer is [tex]\(\boxed{28x^3 + 48x^2 + 62x + 30}\)[/tex], which matches option B.