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To solve the problem, we need to simplify the given expression and match it with one of the provided options. The expression given is:

\[
\frac{x^2 - 2x - 5}{x - 3}
\]

### Step 1: Perform Polynomial Division

We will divide \(x^2 - 2x - 5\) by \(x - 3\).

1. **Divide the first term**: 
   \[
   \frac{x^2}{x} = x
   \]
   Multiply \(x\) by the divisor \(x - 3\):
   \[
   x(x - 3) = x^2 - 3x
   \]

2. **Subtract the result from the original expression**:
   \[
   (x^2 - 2x - 5) - (x^2 - 3x) = (x^2 - 2x - 5) - x^2 + 3x = x - 5
   \]

3. **Now bring down the remainder**:
   \[
   \text{Remaining expression: } x - 5
   \]

Thus, after division, the quotient is \(x\), and the remainder is \(-5\). The expression now looks like:

\[
\frac{x^2 - 2x - 5}{x - 3} = x + \frac{x - 5}{x - 3}
\]

### Step 2: Simplify Further
This doesn't exactly match the options yet. To match the form with the options, we subtract and simplify:

\[
\frac{x - 5}{x - 3} = \frac{(x - 3) - 2}{x - 3} = 1 - \frac{2}{x - 3}
\]

So, the expression becomes:

\[
\frac{x^2 - 2x - 5}{x - 3} = x + 1 - \frac{2}{x - 3}
\]

### Conclusion
The simplified expression matches with Option D:

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

### Final Answer
The correct answer is **D. \(x + 1 - \frac{2}{x - 3}\)**.

Would you like more details or have any further questions?

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### Related Questions:
1. How would the process change if the quadratic had a positive constant term?
2. Can synthetic division be used for this type of problem?
3. What if the divisor were a different linear binomial?
4. How do we factor quadratic expressions in general?
5. Why does subtracting the product from the original polynomial work in division?
6. How does understanding division of polynomials help in solving algebraic fractions?
7. What happens if the numerator polynomial has a higher degree than 2?
8. How can you check if your division result is correct?

### Tip:
When dividing polynomials, always remember to align like terms and ensure you subtract correctly at each step. It’s easy to make sign errors during subtraction!

Learn how to simplify the rational expression (x^2 - 2x - 5) / (x - 3) step-by-step with a detailed solution. This problem covers polynomial division and algebraic concepts, suitable for high school students.

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