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Answer :
To multiply the polynomials [tex]\( (x^2 + 4x + 2) \)[/tex] and [tex]\( (2x^2 + 3x - 4) \)[/tex], we can use the distributive property, which says to multiply each term in the first polynomial by each term in the second polynomial, and then add up all the resulting products. Let's break it down step-by-step:
1. Multiply each term in the first polynomial by 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]\( 4x \cdot 2x^2 = 8x^3 \)[/tex]
- [tex]\( 4x \cdot 3x = 12x^2 \)[/tex]
- [tex]\( 4x \cdot (-4) = -16x \)[/tex]
- [tex]\( 2 \cdot 2x^2 = 4x^2 \)[/tex]
- [tex]\( 2 \cdot 3x = 6x \)[/tex]
- [tex]\( 2 \cdot (-4) = -8 \)[/tex]
2. Add 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:
- 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])
4. Write the simplified expression:
[tex]\[
2x^4 + 11x^3 + 12x^2 - 10x - 8
\][/tex]
So, the product of the polynomials is [tex]\( 2x^4 + 11x^3 + 12x^2 - 10x - 8 \)[/tex].
The correct answer is C: [tex]\( 2x^4 + 11x^3 + 12x^2 - 10x - 8 \)[/tex].
1. Multiply each term in the first polynomial by 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]\( 4x \cdot 2x^2 = 8x^3 \)[/tex]
- [tex]\( 4x \cdot 3x = 12x^2 \)[/tex]
- [tex]\( 4x \cdot (-4) = -16x \)[/tex]
- [tex]\( 2 \cdot 2x^2 = 4x^2 \)[/tex]
- [tex]\( 2 \cdot 3x = 6x \)[/tex]
- [tex]\( 2 \cdot (-4) = -8 \)[/tex]
2. Add 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:
- 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])
4. Write the simplified expression:
[tex]\[
2x^4 + 11x^3 + 12x^2 - 10x - 8
\][/tex]
So, the product of the polynomials is [tex]\( 2x^4 + 11x^3 + 12x^2 - 10x - 8 \)[/tex].
The correct answer is C: [tex]\( 2x^4 + 11x^3 + 12x^2 - 10x - 8 \)[/tex].
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