Answer :
To find the product of the two polynomials [tex]\((x^2 + 4x + 2)\)[/tex] and [tex]\((2x^2 + 3x - 4)\)[/tex], we are going to multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
Here’s how you can do it step-by-step:
1. Multiply each term in the first polynomial by each term in 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 these products together:
[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], we have [tex]\( 2x^4 \)[/tex].
- For [tex]\( x^3 \)[/tex], combine [tex]\( 3x^3 + 8x^3 = 11x^3 \)[/tex].
- For [tex]\( x^2 \)[/tex], combine [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2 \)[/tex].
- For [tex]\( x \)[/tex], combine [tex]\(-16x + 6x = -10x \)[/tex].
- The constant term is [tex]\(-8\)[/tex].
Putting it all together, the final answer is:
[tex]\[ 2x^4 + 11x^3 + 12x^2 - 10x - 8 \][/tex]
This corresponds to choice D.
Here’s how you can do it step-by-step:
1. Multiply each term in the first polynomial by each term in 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 these products together:
[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], we have [tex]\( 2x^4 \)[/tex].
- For [tex]\( x^3 \)[/tex], combine [tex]\( 3x^3 + 8x^3 = 11x^3 \)[/tex].
- For [tex]\( x^2 \)[/tex], combine [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2 \)[/tex].
- For [tex]\( x \)[/tex], combine [tex]\(-16x + 6x = -10x \)[/tex].
- The constant term is [tex]\(-8\)[/tex].
Putting it all together, the final answer is:
[tex]\[ 2x^4 + 11x^3 + 12x^2 - 10x - 8 \][/tex]
This corresponds to choice D.
