Answer :
Sure, let's multiply the two polynomials step-by-step:
We are given:
[tex]\[
(x^2 + 4x + 2) \times (2x^2 + 3x - 4)
\][/tex]
### Step-by-Step Multiplication:
1. 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]
2. Combine all the results:
[tex]\[
2x^4 + 3x^3 - 4x^2 + 8x^3 + 12x^2 - 16x + 4x^2 + 6x - 8
\][/tex]
3. Combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(2x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-16x + 6x = -10x\)[/tex]
- For constant: [tex]\(-8\)[/tex]
Putting it all together, the result is:
[tex]\[
2x^4 + 11x^3 + 12x^2 - 10x - 8
\][/tex]
Thus, the correct answer is B. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
We are given:
[tex]\[
(x^2 + 4x + 2) \times (2x^2 + 3x - 4)
\][/tex]
### Step-by-Step Multiplication:
1. 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]
2. Combine all the results:
[tex]\[
2x^4 + 3x^3 - 4x^2 + 8x^3 + 12x^2 - 16x + 4x^2 + 6x - 8
\][/tex]
3. Combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(2x^4\)[/tex]
- For [tex]\(x^3\)[/tex]: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- For [tex]\(x^2\)[/tex]: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-16x + 6x = -10x\)[/tex]
- For constant: [tex]\(-8\)[/tex]
Putting it all together, the result is:
[tex]\[
2x^4 + 11x^3 + 12x^2 - 10x - 8
\][/tex]
Thus, the correct answer is B. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
