1. **State the problem:** Simplify the expression $$\frac{x^2 - 9}{x^2 - 6x + 9}$$. 2. **Recall the formulas and rules:** - Difference of squares: $$a^2 - b^2 = (a - b)(a + b)$$ - Perfect square trinomial: $$a^2 - 2ab + b^2 = (a - b)^2$$ 3. **Factor numerator:** $$x^2 - 9 = (x - 3)(x + 3)$$ (difference of squares) 4. **Factor denominator:** $$x^2 - 6x + 9 = (x - 3)^2$$ (perfect square trinomial) 5. **Rewrite the expression:** $$\frac{(x - 3)(x + 3)}{(x - 3)^2}$$ 6. **Simplify by canceling common factors:** Cancel one $(x - 3)$ from numerator and denominator: $$\frac{\cancel{(x - 3)}(x + 3)}{\cancel{(x - 3)}(x - 3)} = \frac{x + 3}{x - 3}$$ 7. **State the simplified expression:** $$\frac{x + 3}{x - 3}$$ 8. **Note domain restrictions:** The original denominator cannot be zero, so: $$x - 3 \neq 0 \implies x \neq 3$$ **Final answer:** $$\frac{x + 3}{x - 3}, \quad x \neq 3$$