Answer :
To multiply the polynomials [tex]\((7x^2 + 5x + 7)\)[/tex] and [tex]\((4x - 6)\)[/tex], we'll use the distributive property, also known as the FOIL method for binomials. However, since we are dealing with a trinomial and a binomial, we'll distribute each term in the first polynomial by each term in the second polynomial, following these steps:
### Step-by-Step Solution:
1. Distribute [tex]\(7x^2\)[/tex]:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7x^2 \times 4x = 28x^3
\][/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7x^2 \times (-6) = -42x^2
\][/tex]
2. Distribute [tex]\(5x\)[/tex]:
- Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
5x \times 4x = 20x^2
\][/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
5x \times (-6) = -30x
\][/tex]
3. Distribute [tex]\(7\)[/tex]:
- Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7 \times 4x = 28x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7 \times (-6) = -42
\][/tex]
### Combine Like Terms:
4. Combine like terms:
- [tex]\(x^3\)[/tex] term: [tex]\(28x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(-30x + 28x = -2x\)[/tex]
- Constant term: [tex]\(-42\)[/tex]
### Final Result:
Putting it all together, the result of multiplying the polynomials is:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
Therefore, the correct answer is option D: [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
### Step-by-Step Solution:
1. Distribute [tex]\(7x^2\)[/tex]:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7x^2 \times 4x = 28x^3
\][/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7x^2 \times (-6) = -42x^2
\][/tex]
2. Distribute [tex]\(5x\)[/tex]:
- Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
5x \times 4x = 20x^2
\][/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
5x \times (-6) = -30x
\][/tex]
3. Distribute [tex]\(7\)[/tex]:
- Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7 \times 4x = 28x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7 \times (-6) = -42
\][/tex]
### Combine Like Terms:
4. Combine like terms:
- [tex]\(x^3\)[/tex] term: [tex]\(28x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(-42x^2 + 20x^2 = -22x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(-30x + 28x = -2x\)[/tex]
- Constant term: [tex]\(-42\)[/tex]
### Final Result:
Putting it all together, the result of multiplying the polynomials is:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
Therefore, the correct answer is option D: [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
