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The given problem asks to perform long division on the following expression:

\[
\frac{x^3 + 2x^2 - 3x + 1}{x - 3}
\]

### Steps for Long Division of Polynomials:
1. **Divide the first term of the dividend** by the first term of the divisor. This will give you the first term of the quotient.
   \[
   \frac{x^3}{x} = x^2
   \]
   So, the first term of the quotient is \(x^2\).

2. **Multiply the divisor** by the first term of the quotient.
   \[
   (x - 3) \times x^2 = x^3 - 3x^2
   \]
   Subtract this from the original polynomial.
   \[
   (x^3 + 2x^2 - 3x + 1) - (x^3 - 3x^2) = 5x^2 - 3x + 1
   \]

3. **Divide the first term of the new polynomial** by the first term of the divisor.
   \[
   \frac{5x^2}{x} = 5x
   \]
   So, the next term in the quotient is \(5x\).

4. **Multiply the divisor** by the new term in the quotient.
   \[
   (x - 3) \times 5x = 5x^2 - 15x
   \]
   Subtract this from the current polynomial.
   \[
   (5x^2 - 3x + 1) - (5x^2 - 15x) = 12x + 1
   \]

5. **Divide the first term of the new polynomial** by the first term of the divisor.
   \[
   \frac{12x}{x} = 12
   \]
   So, the next term in the quotient is \(12\).

6. **Multiply the divisor** by this new term.
   \[
   (x - 3) \times 12 = 12x - 36
   \]
   Subtract this from the current polynomial.
   \[
   (12x + 1) - (12x - 36) = 37
   \]

Now, since the degree of the remainder \(37\) is less than the degree of the divisor \(x - 3\), we stop here.

### Final Answer:
- **Quotient**: \(x^2 + 5x + 12\)
- **Remainder**: \(37\)

Would you like a further breakdown or any additional details?

Here are 5 follow-up questions:
1. How do we confirm that the division process is correct?
2. Why is long division important in solving rational expressions?
3. Can synthetic division be used instead of long division in this case? Why or why not?
4. What is the relationship between the degree of the divisor and the stopping point of the division?
5. How does the quotient and remainder help in graphing rational functions?

**Tip**: Always align terms by degree during long division to avoid confusion when subtracting terms.

Perform long division on the polynomial (x^3 + 2x^2 - 3x + 1) ÷ (x - 3). Identify the quotient and remainder.

Master the long division of polynomials with this step-by-step solution for dividing (x^3 + 2x^2 - 3x + 1) by (x - 3). Learn how to find both the quotient (x^2 + 5x + 12) and the remainder (37). Ideal for students in Grades 9-12.

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