Answer :
To properly expand the expression [tex]\((x + 6)(2x^2 + x - 7)\)[/tex], we'll distribute each term in the first polynomial with each term in the second polynomial. Let’s go step-by-step:
1. Distribute [tex]\(x\)[/tex] to each term in [tex]\((2x^2 + x - 7)\)[/tex]:
- [tex]\(x \cdot 2x^2 = 2x^3\)[/tex]
- [tex]\(x \cdot x = x^2\)[/tex]
- [tex]\(x \cdot (-7) = -7x\)[/tex]
Adding these, we get:
[tex]\[
2x^3 + x^2 - 7x
\][/tex]
2. Distribute [tex]\(6\)[/tex] to each term in [tex]\((2x^2 + x - 7)\)[/tex]:
- [tex]\(6 \cdot 2x^2 = 12x^2\)[/tex]
- [tex]\(6 \cdot x = 6x\)[/tex]
- [tex]\(6 \cdot (-7) = -42\)[/tex]
Adding these, we get:
[tex]\[
12x^2 + 6x - 42
\][/tex]
3. Combine all the terms from the two distributions:
[tex]\[
(2x^3 + x^2 - 7x) + (12x^2 + 6x - 42)
\][/tex]
4. Combine like terms:
- For [tex]\(x^3\)[/tex] terms: [tex]\(2x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(x^2 + 12x^2 = 13x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(-7x + 6x = -x\)[/tex]
- Constant term: [tex]\(-42\)[/tex]
Resulting in:
[tex]\[
2x^3 + 13x^2 - x - 42
\][/tex]
The expanded expression matches option D: [tex]\(2x^3 + 13x^2 - x - 42\)[/tex]. So, the correct answer is D.
1. Distribute [tex]\(x\)[/tex] to each term in [tex]\((2x^2 + x - 7)\)[/tex]:
- [tex]\(x \cdot 2x^2 = 2x^3\)[/tex]
- [tex]\(x \cdot x = x^2\)[/tex]
- [tex]\(x \cdot (-7) = -7x\)[/tex]
Adding these, we get:
[tex]\[
2x^3 + x^2 - 7x
\][/tex]
2. Distribute [tex]\(6\)[/tex] to each term in [tex]\((2x^2 + x - 7)\)[/tex]:
- [tex]\(6 \cdot 2x^2 = 12x^2\)[/tex]
- [tex]\(6 \cdot x = 6x\)[/tex]
- [tex]\(6 \cdot (-7) = -42\)[/tex]
Adding these, we get:
[tex]\[
12x^2 + 6x - 42
\][/tex]
3. Combine all the terms from the two distributions:
[tex]\[
(2x^3 + x^2 - 7x) + (12x^2 + 6x - 42)
\][/tex]
4. Combine like terms:
- For [tex]\(x^3\)[/tex] terms: [tex]\(2x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(x^2 + 12x^2 = 13x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(-7x + 6x = -x\)[/tex]
- Constant term: [tex]\(-42\)[/tex]
Resulting in:
[tex]\[
2x^3 + 13x^2 - x - 42
\][/tex]
The expanded expression matches option D: [tex]\(2x^3 + 13x^2 - x - 42\)[/tex]. So, the correct answer is D.
