1. **State the problem:** Simplify the expression $$\frac{X-1}{X-2} + \frac{X-6}{X^2-4}$$. 2. **Identify the formula and rules:** To add rational expressions, find a common denominator. Note that $$X^2-4$$ is a difference of squares and factors as $$(X-2)(X+2)$$. 3. **Rewrite the expression with factored denominators:** $$\frac{X-1}{X-2} + \frac{X-6}{(X-2)(X+2)}$$ 4. **Find the common denominator:** The common denominator is $$(X-2)(X+2)$$. 5. **Rewrite the first fraction to have the common denominator:** $$\frac{X-1}{X-2} = \frac{(X-1)(X+2)}{(X-2)(X+2)}$$ 6. **Add the fractions:** $$\frac{(X-1)(X+2)}{(X-2)(X+2)} + \frac{X-6}{(X-2)(X+2)} = \frac{(X-1)(X+2) + (X-6)}{(X-2)(X+2)}$$ 7. **Expand the numerator:** $$(X-1)(X+2) = X^2 + 2X - X - 2 = X^2 + X - 2$$ 8. **Combine the numerator terms:** $$X^2 + X - 2 + X - 6 = X^2 + 2X - 8$$ 9. **Rewrite the expression:** $$\frac{X^2 + 2X - 8}{(X-2)(X+2)}$$ 10. **Factor the numerator:** $$X^2 + 2X - 8 = (X+4)(X-2)$$ 11. **Simplify the fraction by canceling common factors:** $$\frac{\cancel{(X-2)}(X+4)}{\cancel{(X-2)}(X+2)} = \frac{X+4}{X+2}$$ **Final answer:** $$\boxed{\frac{X+4}{X+2}}$$