Answer :
Let's solve the problem of multiplying the polynomials [tex]\(x^2 + 4x + 2\)[/tex] and [tex]\(2x^2 + 3x - 4\)[/tex] step-by-step.
### Step 1: Distribute Each Term
We'll distribute each term of the first polynomial across the second polynomial.
1. Multiply [tex]\(x^2\)[/tex] by each term in [tex]\(2x^2 + 3x - 4\)[/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 [tex]\(4x\)[/tex] by each term in [tex]\(2x^2 + 3x - 4\)[/tex]:
[tex]\[
4x \times 2x^2 = 8x^3
\][/tex]
[tex]\[
4x \times 3x = 12x^2
\][/tex]
[tex]\[
4x \times (-4) = -16x
\][/tex]
3. Multiply [tex]\(2\)[/tex] by each term in [tex]\(2x^2 + 3x - 4\)[/tex]:
[tex]\[
2 \times 2x^2 = 4x^2
\][/tex]
[tex]\[
2 \times 3x = 6x
\][/tex]
[tex]\[
2 \times (-4) = -8
\][/tex]
### Step 2: Combine Like Terms
Now, let's combine all these results:
- [tex]\(x^4\)[/tex] terms: [tex]\(2x^4\)[/tex]
- [tex]\(x^3\)[/tex] terms: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- Constant term: [tex]\(-8\)[/tex]
Putting it all together, the product is:
[tex]\[
2x^4 + 11x^3 + 12x^2 - 10x - 8
\][/tex]
Therefore, the correct answer is C: [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
### Step 1: Distribute Each Term
We'll distribute each term of the first polynomial across the second polynomial.
1. Multiply [tex]\(x^2\)[/tex] by each term in [tex]\(2x^2 + 3x - 4\)[/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 [tex]\(4x\)[/tex] by each term in [tex]\(2x^2 + 3x - 4\)[/tex]:
[tex]\[
4x \times 2x^2 = 8x^3
\][/tex]
[tex]\[
4x \times 3x = 12x^2
\][/tex]
[tex]\[
4x \times (-4) = -16x
\][/tex]
3. Multiply [tex]\(2\)[/tex] by each term in [tex]\(2x^2 + 3x - 4\)[/tex]:
[tex]\[
2 \times 2x^2 = 4x^2
\][/tex]
[tex]\[
2 \times 3x = 6x
\][/tex]
[tex]\[
2 \times (-4) = -8
\][/tex]
### Step 2: Combine Like Terms
Now, let's combine all these results:
- [tex]\(x^4\)[/tex] terms: [tex]\(2x^4\)[/tex]
- [tex]\(x^3\)[/tex] terms: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- [tex]\(x^2\)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- Constant term: [tex]\(-8\)[/tex]
Putting it all together, the product is:
[tex]\[
2x^4 + 11x^3 + 12x^2 - 10x - 8
\][/tex]
Therefore, the correct answer is C: [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
