Answer :
To multiply the polynomials [tex]\(x^2 + 4x + 2\)[/tex] and [tex]\(2x^2 + 3x - 4\)[/tex], we can use the distributive property (also known as the FOIL method for two binomials) across all terms of the polynomials. Here's a step-by-step explanation:
1. Write down the polynomials:
[tex]\[(x^2 + 4x + 2) \times (2x^2 + 3x - 4)\][/tex]
2. Distribute each term of the first polynomial to each term of the second polynomial:
- Multiply [tex]\(x^2\)[/tex] by each term in the second polynomial:
- [tex]\(x^2 \times 2x^2 = 2x^4\)[/tex]
- [tex]\(x^2 \times 3x = 3x^3\)[/tex]
- [tex]\(x^2 \times -4 = -4x^2\)[/tex]
- Multiply [tex]\(4x\)[/tex] by each term in the second polynomial:
- [tex]\(4x \times 2x^2 = 8x^3\)[/tex]
- [tex]\(4x \times 3x = 12x^2\)[/tex]
- [tex]\(4x \times -4 = -16x\)[/tex]
- Multiply [tex]\(2\)[/tex] by each term in the second polynomial:
- [tex]\(2 \times 2x^2 = 4x^2\)[/tex]
- [tex]\(2 \times 3x = 6x\)[/tex]
- [tex]\(2 \times -4 = -8\)[/tex]
3. Combine all the results:
[tex]\[
2x^4 + 3x^3 - 4x^2 + 8x^3 + 12x^2 - 16x + 4x^2 + 6x - 8
\][/tex]
4. Combine like terms:
- 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]
5. Write the combined expression:
[tex]\[2x^4 + 11x^3 + 12x^2 - 10x - 8\][/tex]
So, the correct answer is [tex]\( \boxed{2x^4 + 11x^3 + 12x^2 - 10x - 8} \)[/tex], which matches option D.
1. Write down the polynomials:
[tex]\[(x^2 + 4x + 2) \times (2x^2 + 3x - 4)\][/tex]
2. Distribute each term of the first polynomial to each term of the second polynomial:
- Multiply [tex]\(x^2\)[/tex] by each term in the second polynomial:
- [tex]\(x^2 \times 2x^2 = 2x^4\)[/tex]
- [tex]\(x^2 \times 3x = 3x^3\)[/tex]
- [tex]\(x^2 \times -4 = -4x^2\)[/tex]
- Multiply [tex]\(4x\)[/tex] by each term in the second polynomial:
- [tex]\(4x \times 2x^2 = 8x^3\)[/tex]
- [tex]\(4x \times 3x = 12x^2\)[/tex]
- [tex]\(4x \times -4 = -16x\)[/tex]
- Multiply [tex]\(2\)[/tex] by each term in the second polynomial:
- [tex]\(2 \times 2x^2 = 4x^2\)[/tex]
- [tex]\(2 \times 3x = 6x\)[/tex]
- [tex]\(2 \times -4 = -8\)[/tex]
3. Combine all the results:
[tex]\[
2x^4 + 3x^3 - 4x^2 + 8x^3 + 12x^2 - 16x + 4x^2 + 6x - 8
\][/tex]
4. Combine like terms:
- 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]
5. Write the combined expression:
[tex]\[2x^4 + 11x^3 + 12x^2 - 10x - 8\][/tex]
So, the correct answer is [tex]\( \boxed{2x^4 + 11x^3 + 12x^2 - 10x - 8} \)[/tex], which matches option D.
