Answer :
To multiply the polynomials [tex]\(x^2 + 4x + 2\)[/tex] and [tex]\(2x^2 + 3x - 4\)[/tex], we need to distribute each term in the first polynomial by each term in the second polynomial and combine like terms afterward.
Here's a step-by-step explanation of the multiplication:
1. Multiply each term by [tex]\(x^2\)[/tex]:
- [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]
2. Multiply each term by [tex]\(4x\)[/tex]:
- [tex]\(4x \times 2x^2 = 8x^3\)[/tex]
- [tex]\(4x \times 3x = 12x^2\)[/tex]
- [tex]\(4x \times (-4) = -16x\)[/tex]
3. Multiply each term by [tex]\(2\)[/tex]:
- [tex]\(2 \times 2x^2 = 4x^2\)[/tex]
- [tex]\(2 \times 3x = 6x\)[/tex]
- [tex]\(2 \times (-4) = -8\)[/tex]
4. Combine all the terms:
- Collect all terms from the products above:
[tex]\[2x^4 + 3x^3 - 4x^2 + 8x^3 + 12x^2 - 16x + 4x^2 + 6x - 8\][/tex]
5. Combine like terms:
- [tex]\(2x^4\)[/tex] (only one [tex]\(x^4\)[/tex])
- [tex]\((3x^3 + 8x^3) = 11x^3\)[/tex]
- [tex]\((-4x^2 + 12x^2 + 4x^2) = 12x^2\)[/tex]
- [tex]\((-16x + 6x) = -10x\)[/tex]
- [tex]\(-8\)[/tex] (only one constant term)
The final result, combining all like terms, is:
[tex]\[2x^4 + 11x^3 + 12x^2 - 10x - 8\][/tex]
This matches with option C. So, the correct answer is C. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
Here's a step-by-step explanation of the multiplication:
1. Multiply each term by [tex]\(x^2\)[/tex]:
- [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]
2. Multiply each term by [tex]\(4x\)[/tex]:
- [tex]\(4x \times 2x^2 = 8x^3\)[/tex]
- [tex]\(4x \times 3x = 12x^2\)[/tex]
- [tex]\(4x \times (-4) = -16x\)[/tex]
3. Multiply each term by [tex]\(2\)[/tex]:
- [tex]\(2 \times 2x^2 = 4x^2\)[/tex]
- [tex]\(2 \times 3x = 6x\)[/tex]
- [tex]\(2 \times (-4) = -8\)[/tex]
4. Combine all the terms:
- Collect all terms from the products above:
[tex]\[2x^4 + 3x^3 - 4x^2 + 8x^3 + 12x^2 - 16x + 4x^2 + 6x - 8\][/tex]
5. Combine like terms:
- [tex]\(2x^4\)[/tex] (only one [tex]\(x^4\)[/tex])
- [tex]\((3x^3 + 8x^3) = 11x^3\)[/tex]
- [tex]\((-4x^2 + 12x^2 + 4x^2) = 12x^2\)[/tex]
- [tex]\((-16x + 6x) = -10x\)[/tex]
- [tex]\(-8\)[/tex] (only one constant term)
The final result, combining all like terms, is:
[tex]\[2x^4 + 11x^3 + 12x^2 - 10x - 8\][/tex]
This matches with option C. So, the correct answer is C. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
