1. **State the problem:** Simplify the expression $$\frac{x-1}{x-2} - \frac{x-2}{x+2}$$. 2. **Formula and rules:** To subtract fractions, find a common denominator and combine the numerators. 3. **Find the common denominator:** The denominators are $x-2$ and $x+2$, so the common denominator is $(x-2)(x+2)$. 4. **Rewrite each fraction with the common denominator:** $$\frac{x-1}{x-2} = \frac{(x-1)(x+2)}{(x-2)(x+2)}$$ $$\frac{x-2}{x+2} = \frac{(x-2)(x-2)}{(x+2)(x-2)}$$ 5. **Subtract the numerators:** $$\frac{(x-1)(x+2) - (x-2)^2}{(x-2)(x+2)}$$ 6. **Expand the numerators:** $$(x-1)(x+2) = x^2 + 2x - x - 2 = x^2 + x - 2$$ $$(x-2)^2 = (x-2)(x-2) = x^2 - 4x + 4$$ 7. **Substitute back:** $$\frac{x^2 + x - 2 - (x^2 - 4x + 4)}{(x-2)(x+2)}$$ 8. **Simplify the numerator:** $$x^2 + x - 2 - x^2 + 4x - 4 = (x^2 - x^2) + (x + 4x) + (-2 - 4) = 5x - 6$$ 9. **Final simplified expression:** $$\frac{5x - 6}{(x-2)(x+2)}$$ 10. **Note:** The denominator can also be written as $x^2 - 4$. **Answer:** $$\frac{5x - 6}{x^2 - 4}$$