Answer :
Sure, let's go through the multiplication of the polynomials step by step.
We need to multiply [tex]\((x^2 + 4x + 2)\)[/tex] by [tex]\((2x^2 + 3x - 4)\)[/tex].
### Step-by-Step Multiplication:
1. Distribute each term in the first polynomial to every term in the second polynomial:
[tex]\[
(x^2 + 4x + 2) \times (2x^2 + 3x - 4)
\][/tex]
2. Multiply [tex]\(x^2\)[/tex] by each term in the second polynomial:
[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 each term in the second polynomial:
[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 each term in the second polynomial:
[tex]\[
2 \cdot 2x^2 = 4x^2
\][/tex]
[tex]\[
2 \cdot 3x = 6x
\][/tex]
[tex]\[
2 \cdot (-4) = -8
\][/tex]
5. Combine all these results:
[tex]\[
2x^4 + 3x^3 - 4x^2 + 8x^3 + 12x^2 - 16x + 4x^2 + 6x - 8
\][/tex]
6. Combine like terms:
- The [tex]\(2x^4\)[/tex] term stands alone.
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- The constant term [tex]\(-8\)[/tex] stands alone.
So, putting it all together, we have:
[tex]\[
2x^4 + 11x^3 + 12x^2 - 10x - 8
\][/tex]
Thus, the final result after multiplying the given polynomials is:
[tex]\[
2 x^4 + 11 x^3 + 12 x^2 - 10 x - 8
\][/tex]
Therefore, the correct answer is:
C. [tex]\(2 x^4 + 11 x^3 + 12 x^2 - 10 x - 8\)[/tex]
We need to multiply [tex]\((x^2 + 4x + 2)\)[/tex] by [tex]\((2x^2 + 3x - 4)\)[/tex].
### Step-by-Step Multiplication:
1. Distribute each term in the first polynomial to every term in the second polynomial:
[tex]\[
(x^2 + 4x + 2) \times (2x^2 + 3x - 4)
\][/tex]
2. Multiply [tex]\(x^2\)[/tex] by each term in the second polynomial:
[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 each term in the second polynomial:
[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 each term in the second polynomial:
[tex]\[
2 \cdot 2x^2 = 4x^2
\][/tex]
[tex]\[
2 \cdot 3x = 6x
\][/tex]
[tex]\[
2 \cdot (-4) = -8
\][/tex]
5. Combine all these results:
[tex]\[
2x^4 + 3x^3 - 4x^2 + 8x^3 + 12x^2 - 16x + 4x^2 + 6x - 8
\][/tex]
6. Combine like terms:
- The [tex]\(2x^4\)[/tex] term stands alone.
- Combine the [tex]\(x^3\)[/tex] terms: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- The constant term [tex]\(-8\)[/tex] stands alone.
So, putting it all together, we have:
[tex]\[
2x^4 + 11x^3 + 12x^2 - 10x - 8
\][/tex]
Thus, the final result after multiplying the given polynomials is:
[tex]\[
2 x^4 + 11 x^3 + 12 x^2 - 10 x - 8
\][/tex]
Therefore, the correct answer is:
C. [tex]\(2 x^4 + 11 x^3 + 12 x^2 - 10 x - 8\)[/tex]
