1. **State the problem:** Simplify the expression $$\frac{x^2 - 4x + 3}{x^2 - 9}$$. 2. **Recall the formulas and rules:** - Factor quadratic expressions when possible. - Use the difference of squares formula: $$a^2 - b^2 = (a - b)(a + b)$$. - Simplify by canceling common factors. 3. **Factor the numerator:** $$x^2 - 4x + 3 = (x - 3)(x - 1)$$ because $$-3 \times -1 = 3$$ and $$-3 + -1 = -4$$. 4. **Factor the denominator:** $$x^2 - 9 = (x - 3)(x + 3)$$ using the difference of squares. 5. **Rewrite the expression with factors:** $$\frac{(x - 3)(x - 1)}{(x - 3)(x + 3)}$$. 6. **Cancel the common factor:** $$\frac{\cancel{(x - 3)}(x - 1)}{\cancel{(x - 3)}(x + 3)} = \frac{x - 1}{x + 3}$$, assuming $$x \neq 3$$ to avoid division by zero. 7. **Final simplified expression:** $$\boxed{\frac{x - 1}{x + 3}}$$. This is the simplified form of the original expression, valid for all $$x$$ except $$x = 3$$ and $$x = -3$$ where the original expression is undefined.