Answer :
To multiply the polynomials [tex]\( (x^2 + 4x + 2) \)[/tex] and [tex]\( (2x^2 + 3x - 4) \)[/tex], follow these steps:
1. Distribute Each Term:
You'll distribute each term in the first polynomial by each term in the second polynomial.
2. Multiply Terms:
- First, 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]
- Next, 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]
- Finally, 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]
3. Combine Like Terms:
Now add all the products together and combine the like terms:
- From [tex]\( 2x^4 \)[/tex], we only have [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].
So the result is:
[tex]\[ 2x^4 + 11x^3 + 12x^2 - 10x - 8 \][/tex]
The correct answer is option A: [tex]\( 2x^4 + 11x^3 + 12x^2 - 10x - 8 \)[/tex].
1. Distribute Each Term:
You'll distribute each term in the first polynomial by each term in the second polynomial.
2. Multiply Terms:
- First, 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]
- Next, 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]
- Finally, 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]
3. Combine Like Terms:
Now add all the products together and combine the like terms:
- From [tex]\( 2x^4 \)[/tex], we only have [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].
So the result is:
[tex]\[ 2x^4 + 11x^3 + 12x^2 - 10x - 8 \][/tex]
The correct answer is option A: [tex]\( 2x^4 + 11x^3 + 12x^2 - 10x - 8 \)[/tex].
