Answer :
To multiply the polynomials [tex]\((x^2 + 4x + 2)\)[/tex] and [tex]\((2x^2 + 3x - 4)\)[/tex], we'll go through the following steps:
1. Distribute Each Term:
- First, distribute each term of the first polynomial over the entire second polynomial.
2. Multiply Each Part:
- Multiply [tex]\(x^2\)[/tex] by each term of 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 of 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 of 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:
Combine all terms, particularly focusing on combining those with the same powers of [tex]\(x\)[/tex].
- 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]
4. Write the Final Polynomial:
After simplifying and combining like terms, the resulting polynomial is:
[tex]\[
2x^4 + 11x^3 + 12x^2 - 10x - 8
\][/tex]
Now, comparing with the provided options, the correct choice is B. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
1. Distribute Each Term:
- First, distribute each term of the first polynomial over the entire second polynomial.
2. Multiply Each Part:
- Multiply [tex]\(x^2\)[/tex] by each term of 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 of 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 of 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:
Combine all terms, particularly focusing on combining those with the same powers of [tex]\(x\)[/tex].
- 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]
4. Write the Final Polynomial:
After simplifying and combining like terms, the resulting polynomial is:
[tex]\[
2x^4 + 11x^3 + 12x^2 - 10x - 8
\][/tex]
Now, comparing with the provided options, the correct choice is B. [tex]\(2x^4 + 11x^3 + 12x^2 - 10x - 8\)[/tex].
