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To solve this problem, we’ll use synthetic division to divide the polynomial \(2x^3 + 5x^2 - x + 2\) by \(x - 2\).

### Step-by-Step Synthetic Division:

1. **Set up the synthetic division**: 
   - The divisor is \(x - 2\), so we use \(2\) as the number outside the division.
   - The coefficients of the polynomial \(2x^3 + 5x^2 - x + 2\) are \(2\), \(5\), \(-1\), and \(2\).

2. **Perform the synthetic division**:
   - Write down the coefficients: \(2, 5, -1, 2\).
   - Bring down the first coefficient (2).
   - Multiply \(2\) (from outside) by \(2\) (first coefficient), which gives \(4\). Write this under the second coefficient.
   - Add \(5 + 4 = 9\).
   - Multiply \(2\) (outside) by \(9\), giving \(18\). Write this under the next coefficient.
   - Add \(-1 + 18 = 17\).
   - Multiply \(2\) by \(17\), giving \(34\). Write this under the last coefficient.
   - Add \(2 + 34 = 36\).

3. **Result**:
   - The last number, \(36\), is the remainder.
   - The other numbers, \(2\), \(9\), and \(17\), represent the coefficients of the quotient polynomial.

### Final Answer:
- **Quotient**: \(2x^2 + 9x + 17\)
- **Remainder**: \(36\)

Would you like more details on synthetic division, or have any other questions?

### Related Questions to Explore:
1. How do we handle synthetic division if the divisor has a leading coefficient other than 1?
2. What is the connection between synthetic division and the Remainder Theorem?
3. How can synthetic division be used to find roots of a polynomial?
4. What are the advantages of synthetic division over long division in polynomial division?
5. How does synthetic division help in factorizing polynomials?

### Quick Tip:
Always check that all terms in the polynomial are accounted for (including any zero coefficients) before starting synthetic division.

Use synthetic division to find the quotient and remainder when 2x^3 + 5x^2 - x + 2 is divided by x - 2.

Learn how to use synthetic division to find the quotient and remainder when dividing the polynomial 2x^3 + 5x^2 - x + 2 by x - 2. This example demonstrates key polynomial concepts and the Remainder Theorem, suitable for high school students in Grades 10-12.

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