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Answer :
To multiply the polynomials [tex]\( x^2 + 4x + 2 \)[/tex] and [tex]\( 2x^2 + 3x - 4 \)[/tex], we'll use the distributive property, multiplying each term in the first polynomial by each term in the second polynomial, and then combining like terms.
Here's the step-by-step procedure:
1. Multiply each term in [tex]\( x^2 + 4x + 2 \)[/tex] by each term in [tex]\( 2x^2 + 3x - 4 \)[/tex]:
- [tex]\( x^2 \)[/tex] times each term in the second polynomial:
- [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]
- [tex]\( 4x \)[/tex] times each term in the second polynomial:
- [tex]\( 4x \cdot 2x^2 = 8x^3 \)[/tex]
- [tex]\( 4x \cdot 3x = 12x^2 \)[/tex]
- [tex]\( 4x \cdot (-4) = -16x \)[/tex]
- [tex]\( 2 \)[/tex] times each term in the second polynomial:
- [tex]\( 2 \cdot 2x^2 = 4x^2 \)[/tex]
- [tex]\( 2 \cdot 3x = 6x \)[/tex]
- [tex]\( 2 \cdot (-4) = -8 \)[/tex]
2. Combine like terms:
- Combine the [tex]\( x^4 \)[/tex] terms: [tex]\( 2x^4 \)[/tex]
- Combine the [tex]\( x^3 \)[/tex] terms: [tex]\( 3x^3 + 8x^3 = 11x^3 \)[/tex]
- Combine the [tex]\( x^2 \)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- Combine the [tex]\( x \)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- Combine the constant terms: [tex]\(-8\)[/tex]
3. Final polynomial expression:
[tex]\[
2x^4 + 11x^3 + 12x^2 - 10x - 8
\][/tex]
Therefore, the correct answer is B. [tex]\( 2x^4 + 11x^3 + 12x^2 - 10x - 8 \)[/tex].
Here's the step-by-step procedure:
1. Multiply each term in [tex]\( x^2 + 4x + 2 \)[/tex] by each term in [tex]\( 2x^2 + 3x - 4 \)[/tex]:
- [tex]\( x^2 \)[/tex] times each term in the second polynomial:
- [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]
- [tex]\( 4x \)[/tex] times each term in the second polynomial:
- [tex]\( 4x \cdot 2x^2 = 8x^3 \)[/tex]
- [tex]\( 4x \cdot 3x = 12x^2 \)[/tex]
- [tex]\( 4x \cdot (-4) = -16x \)[/tex]
- [tex]\( 2 \)[/tex] times each term in the second polynomial:
- [tex]\( 2 \cdot 2x^2 = 4x^2 \)[/tex]
- [tex]\( 2 \cdot 3x = 6x \)[/tex]
- [tex]\( 2 \cdot (-4) = -8 \)[/tex]
2. Combine like terms:
- Combine the [tex]\( x^4 \)[/tex] terms: [tex]\( 2x^4 \)[/tex]
- Combine the [tex]\( x^3 \)[/tex] terms: [tex]\( 3x^3 + 8x^3 = 11x^3 \)[/tex]
- Combine the [tex]\( x^2 \)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- Combine the [tex]\( x \)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- Combine the constant terms: [tex]\(-8\)[/tex]
3. Final polynomial expression:
[tex]\[
2x^4 + 11x^3 + 12x^2 - 10x - 8
\][/tex]
Therefore, the correct answer is B. [tex]\( 2x^4 + 11x^3 + 12x^2 - 10x - 8 \)[/tex].
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