Answer :
To multiply the polynomials [tex]\((x^2 + 4x + 2)\)[/tex] and [tex]\((2x^2 + 3x - 4)\)[/tex], we will use the distributive property of multiplication for polynomials. Here’s how you can do it step-by-step:
1. Multiply each term of the first polynomial by each term of the second polynomial:
Let's break it down into parts:
- Multiply [tex]\(x^2\)[/tex] with each term of [tex]\((2x^2 + 3x - 4)\)[/tex]:
- [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]
- Multiply [tex]\(4x\)[/tex] with each term of [tex]\((2x^2 + 3x - 4)\)[/tex]:
- [tex]\(4x \cdot 2x^2 = 8x^3\)[/tex]
- [tex]\(4x \cdot 3x = 12x^2\)[/tex]
- [tex]\(4x \cdot (-4) = -16x\)[/tex]
- Multiply [tex]\(2\)[/tex] with each term of [tex]\((2x^2 + 3x - 4)\)[/tex]:
- [tex]\(2 \cdot 2x^2 = 4x^2\)[/tex]
- [tex]\(2 \cdot 3x = 6x\)[/tex]
- [tex]\(2 \cdot (-4) = -8\)[/tex]
2. Combine all the terms obtained from the multiplication:
[tex]\[ 2x^4 + 3x^3 - 4x^2 + 8x^3 + 12x^2 - 16x + 4x^2 + 6x - 8 \][/tex]
3. Combine like terms:
- The terms for [tex]\(x^4\)[/tex] are: [tex]\(2x^4\)[/tex]
- The terms for [tex]\(x^3\)[/tex] are: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- The terms for [tex]\(x^2\)[/tex] are: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- The terms for [tex]\(x\)[/tex] are: [tex]\(-16x + 6x = -10x\)[/tex]
- The constant term is: [tex]\(-8\)[/tex]
Combining all these, the final polynomial is:
[tex]\[ 2x^4 + 11x^3 + 12x^2 - 10x - 8 \][/tex]
Thus, the correct answer is:
C. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex]
1. Multiply each term of the first polynomial by each term of the second polynomial:
Let's break it down into parts:
- Multiply [tex]\(x^2\)[/tex] with each term of [tex]\((2x^2 + 3x - 4)\)[/tex]:
- [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]
- Multiply [tex]\(4x\)[/tex] with each term of [tex]\((2x^2 + 3x - 4)\)[/tex]:
- [tex]\(4x \cdot 2x^2 = 8x^3\)[/tex]
- [tex]\(4x \cdot 3x = 12x^2\)[/tex]
- [tex]\(4x \cdot (-4) = -16x\)[/tex]
- Multiply [tex]\(2\)[/tex] with each term of [tex]\((2x^2 + 3x - 4)\)[/tex]:
- [tex]\(2 \cdot 2x^2 = 4x^2\)[/tex]
- [tex]\(2 \cdot 3x = 6x\)[/tex]
- [tex]\(2 \cdot (-4) = -8\)[/tex]
2. Combine all the terms obtained from the multiplication:
[tex]\[ 2x^4 + 3x^3 - 4x^2 + 8x^3 + 12x^2 - 16x + 4x^2 + 6x - 8 \][/tex]
3. Combine like terms:
- The terms for [tex]\(x^4\)[/tex] are: [tex]\(2x^4\)[/tex]
- The terms for [tex]\(x^3\)[/tex] are: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- The terms for [tex]\(x^2\)[/tex] are: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- The terms for [tex]\(x\)[/tex] are: [tex]\(-16x + 6x = -10x\)[/tex]
- The constant term is: [tex]\(-8\)[/tex]
Combining all these, the final polynomial is:
[tex]\[ 2x^4 + 11x^3 + 12x^2 - 10x - 8 \][/tex]
Thus, the correct answer is:
C. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex]
