Answer :
To multiply the polynomials [tex]\(x^2 + 4x + 2\)[/tex] and [tex]\(2x^2 + 3x - 4\)[/tex], we'll use the distributive property, often called the FOIL method for binomials. However, since these are not binomials, we'll essentially apply the distributive property to each term.
Here's a step-by-step breakdown:
1. Distribute each term of the first polynomial: We'll take each term in [tex]\(x^2 + 4x + 2\)[/tex] and multiply it by every term in [tex]\(2x^2 + 3x - 4\)[/tex].
2. Multiply [tex]\(x^2\)[/tex] by [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]
3. Multiply [tex]\(4x\)[/tex] by [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]
4. Multiply [tex]\(2\)[/tex] by [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]
5. Combine like terms:
- Combine all the terms to collect like powers of [tex]\(x\)[/tex]:
- [tex]\(2x^4 + 3x^3 + 8x^3 - 4x^2 + 12x^2 + 4x^2 - 16x + 6x - 8\)[/tex]
Now, add up the coefficients for terms with the same degree:
- [tex]\(2x^4\)[/tex] (no other [tex]\(x^4\)[/tex] terms)
- Combine [tex]\(3x^3\)[/tex] and [tex]\(8x^3\)[/tex] to get [tex]\(11x^3\)[/tex]
- Combine [tex]\(-4x^2\)[/tex], [tex]\(12x^2\)[/tex], and [tex]\(4x^2\)[/tex] to get [tex]\(12x^2\)[/tex]
- Combine [tex]\(-16x\)[/tex] and [tex]\(6x\)[/tex] to get [tex]\(-10x\)[/tex]
- The constant term is [tex]\(-8\)[/tex]
The final result of multiplying the two polynomials is:
[tex]\[2x^4 + 11x^3 + 12x^2 - 10x - 8\][/tex]
The correct answer is [tex]\(\boxed{C}\)[/tex].
Here's a step-by-step breakdown:
1. Distribute each term of the first polynomial: We'll take each term in [tex]\(x^2 + 4x + 2\)[/tex] and multiply it by every term in [tex]\(2x^2 + 3x - 4\)[/tex].
2. Multiply [tex]\(x^2\)[/tex] by [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]
3. Multiply [tex]\(4x\)[/tex] by [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]
4. Multiply [tex]\(2\)[/tex] by [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]
5. Combine like terms:
- Combine all the terms to collect like powers of [tex]\(x\)[/tex]:
- [tex]\(2x^4 + 3x^3 + 8x^3 - 4x^2 + 12x^2 + 4x^2 - 16x + 6x - 8\)[/tex]
Now, add up the coefficients for terms with the same degree:
- [tex]\(2x^4\)[/tex] (no other [tex]\(x^4\)[/tex] terms)
- Combine [tex]\(3x^3\)[/tex] and [tex]\(8x^3\)[/tex] to get [tex]\(11x^3\)[/tex]
- Combine [tex]\(-4x^2\)[/tex], [tex]\(12x^2\)[/tex], and [tex]\(4x^2\)[/tex] to get [tex]\(12x^2\)[/tex]
- Combine [tex]\(-16x\)[/tex] and [tex]\(6x\)[/tex] to get [tex]\(-10x\)[/tex]
- The constant term is [tex]\(-8\)[/tex]
The final result of multiplying the two polynomials is:
[tex]\[2x^4 + 11x^3 + 12x^2 - 10x - 8\][/tex]
The correct answer is [tex]\(\boxed{C}\)[/tex].
