Answer :
Certainly! Let's go through the problem together.
We need to expand the expression [tex]\((x + 6)(2x^2 + x - 7)\)[/tex] and match it with one of the given options.
### Step-by-step Expansion:
1. Distribute [tex]\(x\)[/tex] through [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]
2. Distribute [tex]\(6\)[/tex] through [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]
3. Combine all terms:
- [tex]\(2x^3\)[/tex] (from the distribution of [tex]\(x\)[/tex])
- [tex]\(x^2 + 12x^2 = 13x^2\)[/tex]
- [tex]\(-7x + 6x = -x\)[/tex]
- [tex]\(-42\)[/tex]
Putting it all together, the expanded expression is:
[tex]\[2x^3 + 13x^2 - x - 42\][/tex]
### Matching with the Options:
Let's look at the given options:
- A: [tex]\(14x^3 + 31x^2 + 8x - 5\)[/tex]
- B: [tex]\(6x^3 - 15x^2 - 1\)[/tex]
- C: [tex]\(42x^3 - 19x^2 - 56x - 15\)[/tex]
- D: [tex]\(2x^3 + 13x^2 - x - 42\)[/tex]
Comparing our expanded expression [tex]\(2x^3 + 13x^2 - x - 42\)[/tex] with the options, it matches Option D.
So, the correct answer is Option D: [tex]\(2x^3 + 13x^2 - x - 42\)[/tex].
We need to expand the expression [tex]\((x + 6)(2x^2 + x - 7)\)[/tex] and match it with one of the given options.
### Step-by-step Expansion:
1. Distribute [tex]\(x\)[/tex] through [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]
2. Distribute [tex]\(6\)[/tex] through [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]
3. Combine all terms:
- [tex]\(2x^3\)[/tex] (from the distribution of [tex]\(x\)[/tex])
- [tex]\(x^2 + 12x^2 = 13x^2\)[/tex]
- [tex]\(-7x + 6x = -x\)[/tex]
- [tex]\(-42\)[/tex]
Putting it all together, the expanded expression is:
[tex]\[2x^3 + 13x^2 - x - 42\][/tex]
### Matching with the Options:
Let's look at the given options:
- A: [tex]\(14x^3 + 31x^2 + 8x - 5\)[/tex]
- B: [tex]\(6x^3 - 15x^2 - 1\)[/tex]
- C: [tex]\(42x^3 - 19x^2 - 56x - 15\)[/tex]
- D: [tex]\(2x^3 + 13x^2 - x - 42\)[/tex]
Comparing our expanded expression [tex]\(2x^3 + 13x^2 - x - 42\)[/tex] with the options, it matches Option D.
So, the correct answer is Option D: [tex]\(2x^3 + 13x^2 - x - 42\)[/tex].
