Question: Simplify the rational expression by canceling common factors:
$$\frac{x^2 - 9x + 18}{x^2 - x - 30}$$
Answer choices:
$$\frac{x - 3}{x + 5}$$
$$\frac{x + 5}{x - 3}$$
$$\frac{x - 6}{x - 3}$$
$$\frac{x - 6}{x + 5}$$
1. **State the problem:** Simplify the rational expression $$\frac{x^2 - 9x + 18}{x^2 - x - 30}$$ by factoring and canceling common factors.
2. **Factor numerator:**
$$x^2 - 9x + 18 = (x - 6)(x - 3)$$
3. **Factor denominator:**
$$x^2 - x - 30 = (x - 6)(x + 5)$$
4. **Rewrite the expression:**
$$\frac{(x - 6)(x - 3)}{(x - 6)(x + 5)}$$
5. **Cancel common factor:**
$$\frac{\cancel{(x - 6)}(x - 3)}{\cancel{(x - 6)}(x + 5)} = \frac{x - 3}{x + 5}$$
6. **Final simplified form:**
$$\frac{x - 3}{x + 5}$$
7. **Answer:** The simplified expression matches the first answer choice $$\frac{x - 3}{x + 5}$$.