Answer :
To multiply the given polynomials [tex]\(x^2 + 4x + 2\)[/tex] and [tex]\(2x^2 + 3x - 4\)[/tex], we will apply the distributive property, which involves multiplying each term from the first polynomial by each term of the second polynomial.
Let's break it down step by step:
1. Distribute each term of the first 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 the terms:
Now, we add all the products together:
[tex]\[
2x^4 + 3x^3 - 4x^2 + 8x^3 + 12x^2 - 16x + 4x^2 + 6x - 8
\][/tex]
3. Simplify by combining like terms:
- The [tex]\(x^4\)[/tex] term: [tex]\(2x^4\)[/tex]
- The [tex]\(x^3\)[/tex] terms: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- The constant term: [tex]\(-8\)[/tex]
Putting it all together, the product is:
[tex]\[ 2x^4 + 11x^3 + 12x^2 - 10x - 8 \][/tex]
Therefore, the correct answer is D. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
Let's break it down step by step:
1. Distribute each term of the first 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 the terms:
Now, we add all the products together:
[tex]\[
2x^4 + 3x^3 - 4x^2 + 8x^3 + 12x^2 - 16x + 4x^2 + 6x - 8
\][/tex]
3. Simplify by combining like terms:
- The [tex]\(x^4\)[/tex] term: [tex]\(2x^4\)[/tex]
- The [tex]\(x^3\)[/tex] terms: [tex]\(3x^3 + 8x^3 = 11x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-4x^2 + 12x^2 + 4x^2 = 12x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-16x + 6x = -10x\)[/tex]
- The constant term: [tex]\(-8\)[/tex]
Putting it all together, the product is:
[tex]\[ 2x^4 + 11x^3 + 12x^2 - 10x - 8 \][/tex]
Therefore, the correct answer is D. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
