To find the quotient when dividing \( P(x) = x^3 - 4x^2 + 2x - 5 \) by \( x - 2 \), we can use polynomial long division.

### Step-by-Step Polynomial Long Division:

1. **Set up the division:**
   \[
   \frac{x^3 - 4x^2 + 2x - 5}{x - 2}
   \]

2. **Divide the first term:**
   - Divide \( x^3 \) by \( x \): \( x^2 \).
   - Multiply \( x^2 \) by the divisor \( x - 2 \): \( x^3 - 2x^2 \).
   - Subtract this from the original polynomial:
     \[
     (x^3 - 4x^2 + 2x - 5) - (x^3 - 2x^2) = -2x^2 + 2x - 5
     \]

3. **Divide the next term:**
   - Divide \( -2x^2 \) by \( x \): \( -2x \).
   - Multiply \( -2x \) by the divisor \( x - 2 \): \( -2x^2 + 4x \).
   - Subtract this from the previous result:
     \[
     (-2x^2 + 2x - 5) - (-2x^2 + 4x) = -2x - 5
     \]

4. **Divide the final term:**
   - Divide \( -2x \) by \( x \): \( -2 \).
   - Multiply \( -2 \) by the divisor \( x - 2 \): \( -2x + 4 \).
   - Subtract this from the previous result:
     \[
     (-2x - 5) - (-2x + 4) = -9
     \]

### Result:
- **Quotient**: \( x^2 - 2x - 2 \)
- **Remainder**: \( -9 \)

Thus, the division can be expressed as:
\[
P(x) = (x - 2)(x^2 - 2x - 2) + (-9)
\]

Would you like any further details on the process or another question answered?

### Related Questions:
1. How can synthetic division be used as an alternative to polynomial long division?
2. What is the significance of the quotient in polynomial division?
3. How do you factor polynomials after using division?
4. Can the remainder theorem help in identifying rational roots?
5. How does polynomial division relate to dividing numbers?

### Tip:
Understanding polynomial long division is essential because it forms the foundation for more advanced algebraic techniques, such as partial fraction decomposition and solving polynomial equations.

Explore how to divide the polynomial P(x) = x^3 - 4x^2 + 2x - 5 by x - 2 using polynomial long division. Understand the process, find the quotient and remainder, and grasp key algebraic concepts. Suitable for advanced high school students.

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