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------------------------------------------------ Multiply the following polynomials:

[tex]
\left(3x^2 - 4x + 5\right)\left(x^2 - 3x + 2\right)
[/tex]

Choose the correct answer:

A. [tex]3x^4 + 12x^2 + 10[/tex]

B. [tex]3x^4 + 10x^2 + 12x + 10[/tex]

C. [tex]3x^4 - 13x^3 + 23x^2 - 23x + 10[/tex]

D. [tex]4x^2 - 7x + 7[/tex]

To multiply the polynomials [tex]\((3x^2 - 4x + 5)\)[/tex] and [tex]\((x^2 - 3x + 2)\)[/tex], we use the distributive property, also known as the FOIL method for binomials.

Let's multiply each term in the first polynomial by each term in the second polynomial:

1. First Term: [tex]\(3x^2 \cdot x^2 = 3x^4\)[/tex]

2. Outer Term: [tex]\(3x^2 \cdot (-3x) = -9x^3\)[/tex]

3. Inner Term: [tex]\(3x^2 \cdot 2 = 6x^2\)[/tex]

4. Second Term: [tex]\(-4x \cdot x^2 = -4x^3\)[/tex]

5. Outer Term: [tex]\(-4x \cdot (-3x) = 12x^2\)[/tex]

6. Inner Term: [tex]\(-4x \cdot 2 = -8x\)[/tex]

7. Third Term: [tex]\(5 \cdot x^2 = 5x^2\)[/tex]

8. Outer Term: [tex]\(5 \cdot (-3x) = -15x\)[/tex]

9. Inner Term: [tex]\(5 \cdot 2 = 10\)[/tex]

Now, let's add all these results together:

- Combine all the terms:
- [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]

The final result is:

[tex]\[3x^4 - 13x^3 + 23x^2 - 23x + 10\][/tex]

Thus, the correct option is:

C. [tex]\(3x^4 - 13x^3 + 23x^2 - 23x + 10\)[/tex]