Answer :
Sure! Let's walk through the process of multiplying the polynomials step by step to find the correct answer.
We want to multiply the two polynomials:
1. [tex]\( x^2 + 4x + 2 \)[/tex]
2. [tex]\( 2x^2 + 3x - 4 \)[/tex]
We'll multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
### Steps:
1. Multiply the first term of the first polynomial [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]
2. Multiply the second term of the first polynomial [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]
3. Multiply the third term of the first polynomial [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]
### Combine all the results:
Now, let's add together all the terms obtained from each step:
- For [tex]\(x^4\)[/tex] terms: [tex]\(2x^4\)[/tex]
- For [tex]\(x^3\)[/tex] terms: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- Constant term: [tex]\(-8\)[/tex]
Final polynomial:
[tex]\[ 2x^4 + 11x^3 + 12x^2 - 10x - 8 \][/tex]
So, the correct answer is D. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
We want to multiply the two polynomials:
1. [tex]\( x^2 + 4x + 2 \)[/tex]
2. [tex]\( 2x^2 + 3x - 4 \)[/tex]
We'll multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
### Steps:
1. Multiply the first term of the first polynomial [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]
2. Multiply the second term of the first polynomial [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]
3. Multiply the third term of the first polynomial [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]
### Combine all the results:
Now, let's add together all the terms obtained from each step:
- For [tex]\(x^4\)[/tex] terms: [tex]\(2x^4\)[/tex]
- For [tex]\(x^3\)[/tex] terms: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- Constant term: [tex]\(-8\)[/tex]
Final polynomial:
[tex]\[ 2x^4 + 11x^3 + 12x^2 - 10x - 8 \][/tex]
So, the correct answer is D. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
