Answer :
To solve the problem, we need to multiply the two polynomials: [tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex]. Let's go through the multiplication step by step.
First, distribute each term of the first polynomial to each term of the second polynomial:
1. Multiply [tex]\(3x^2\)[/tex] by each term in the second polynomial [tex]\((x^2 - 3x + 2)\)[/tex]:
- [tex]\(3x^2 \cdot x^2 = 3x^4\)[/tex]
- [tex]\(3x^2 \cdot (-3x) = -9x^3\)[/tex]
- [tex]\(3x^2 \cdot 2 = 6x^2\)[/tex]
2. Multiply [tex]\(-4x\)[/tex] by each term in the second polynomial:
- [tex]\(-4x \cdot x^2 = -4x^3\)[/tex]
- [tex]\(-4x \cdot (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \cdot 2 = -8x\)[/tex]
3. Multiply [tex]\(5\)[/tex] by each term in the second polynomial:
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot (-3x) = -15x\)[/tex]
- [tex]\(5 \cdot 2 = 10\)[/tex]
Now, combine all the terms obtained from above:
- [tex]\(3x^4\)[/tex]
- [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- [tex]\(-8x - 15x = -23x\)[/tex]
- [tex]\(10\)[/tex]
Putting all of these together, the expanded form of the polynomial is:
[tex]\[3x^4 - 13x^3 + 23x^2 - 23x + 10\][/tex]
Therefore, the correct answer is [tex]\( \boxed{3x^4 - 13x^3 + 23x^2 - 23x + 10} \)[/tex], which corresponds to option A.
First, distribute each term of the first polynomial to each term of the second polynomial:
1. Multiply [tex]\(3x^2\)[/tex] by each term in the second polynomial [tex]\((x^2 - 3x + 2)\)[/tex]:
- [tex]\(3x^2 \cdot x^2 = 3x^4\)[/tex]
- [tex]\(3x^2 \cdot (-3x) = -9x^3\)[/tex]
- [tex]\(3x^2 \cdot 2 = 6x^2\)[/tex]
2. Multiply [tex]\(-4x\)[/tex] by each term in the second polynomial:
- [tex]\(-4x \cdot x^2 = -4x^3\)[/tex]
- [tex]\(-4x \cdot (-3x) = 12x^2\)[/tex]
- [tex]\(-4x \cdot 2 = -8x\)[/tex]
3. Multiply [tex]\(5\)[/tex] by each term in the second polynomial:
- [tex]\(5 \cdot x^2 = 5x^2\)[/tex]
- [tex]\(5 \cdot (-3x) = -15x\)[/tex]
- [tex]\(5 \cdot 2 = 10\)[/tex]
Now, combine all the terms obtained from above:
- [tex]\(3x^4\)[/tex]
- [tex]\(-9x^3 - 4x^3 = -13x^3\)[/tex]
- [tex]\(6x^2 + 12x^2 + 5x^2 = 23x^2\)[/tex]
- [tex]\(-8x - 15x = -23x\)[/tex]
- [tex]\(10\)[/tex]
Putting all of these together, the expanded form of the polynomial is:
[tex]\[3x^4 - 13x^3 + 23x^2 - 23x + 10\][/tex]
Therefore, the correct answer is [tex]\( \boxed{3x^4 - 13x^3 + 23x^2 - 23x + 10} \)[/tex], which corresponds to option A.
