Answer :
To multiply the polynomials [tex]\(x^2 + 4x + 2\)[/tex] and [tex]\(2x^2 + 3x - 4\)[/tex], follow these steps:
1. Distribute Each Term: Multiply each term in the first polynomial by each term in the second polynomial.
2. Multiply Terms:
- [tex]\(x^2\)[/tex] times each term in [tex]\(2x^2 + 3x - 4\)[/tex]:
- [tex]\(x^2 \cdot 2x^2 = 2x^4\)[/tex]
- [tex]\(x^2 \cdot 3x = 3x^3\)[/tex]
- [tex]\(x^2 \cdot (-4) = -4x^2\)[/tex]
- [tex]\(4x\)[/tex] times each term in [tex]\(2x^2 + 3x - 4\)[/tex]:
- [tex]\(4x \cdot 2x^2 = 8x^3\)[/tex]
- [tex]\(4x \cdot 3x = 12x^2\)[/tex]
- [tex]\(4x \cdot (-4) = -16x\)[/tex]
- [tex]\(2\)[/tex] times each term in [tex]\(2x^2 + 3x - 4\)[/tex]:
- [tex]\(2 \cdot 2x^2 = 4x^2\)[/tex]
- [tex]\(2 \cdot 3x = 6x\)[/tex]
- [tex]\(2 \cdot (-4) = -8\)[/tex]
3. Combine Like Terms:
- Combine all the terms from the multiplication:
- Combine [tex]\(x^4\)[/tex] terms: [tex]\(2x^4\)[/tex]
- Combine [tex]\(x^3\)[/tex] terms: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- Combine [tex]\(x\)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- Combine constant terms: [tex]\(-8\)[/tex]
So, the final result is:
[tex]\[2x^4 + 11x^3 + 12x^2 - 10x - 8\][/tex]
Therefore, the correct answer is option D: [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
1. Distribute Each Term: Multiply each term in the first polynomial by each term in the second polynomial.
2. Multiply Terms:
- [tex]\(x^2\)[/tex] times each term in [tex]\(2x^2 + 3x - 4\)[/tex]:
- [tex]\(x^2 \cdot 2x^2 = 2x^4\)[/tex]
- [tex]\(x^2 \cdot 3x = 3x^3\)[/tex]
- [tex]\(x^2 \cdot (-4) = -4x^2\)[/tex]
- [tex]\(4x\)[/tex] times each term in [tex]\(2x^2 + 3x - 4\)[/tex]:
- [tex]\(4x \cdot 2x^2 = 8x^3\)[/tex]
- [tex]\(4x \cdot 3x = 12x^2\)[/tex]
- [tex]\(4x \cdot (-4) = -16x\)[/tex]
- [tex]\(2\)[/tex] times each term in [tex]\(2x^2 + 3x - 4\)[/tex]:
- [tex]\(2 \cdot 2x^2 = 4x^2\)[/tex]
- [tex]\(2 \cdot 3x = 6x\)[/tex]
- [tex]\(2 \cdot (-4) = -8\)[/tex]
3. Combine Like Terms:
- Combine all the terms from the multiplication:
- Combine [tex]\(x^4\)[/tex] terms: [tex]\(2x^4\)[/tex]
- Combine [tex]\(x^3\)[/tex] terms: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- Combine [tex]\(x\)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- Combine constant terms: [tex]\(-8\)[/tex]
So, the final result is:
[tex]\[2x^4 + 11x^3 + 12x^2 - 10x - 8\][/tex]
Therefore, the correct answer is option D: [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
