1. **State the problem:** Simplify the expression $$\frac{x - 2}{x + 2} + \frac{x + 10}{x^2 + 6x + 8}$$. 2. **Factor the denominator:** Notice that $$x^2 + 6x + 8$$ can be factored as $$(x + 2)(x + 4)$$. 3. **Rewrite the expression:** $$\frac{x - 2}{x + 2} + \frac{x + 10}{(x + 2)(x + 4)}$$ 4. **Find a common denominator:** The common denominator is $$(x + 2)(x + 4)$$. 5. **Rewrite the first fraction with the common denominator:** $$\frac{(x - 2)(x + 4)}{(x + 2)(x + 4)} + \frac{x + 10}{(x + 2)(x + 4)}$$ 6. **Expand the numerator of the first fraction:** $$(x - 2)(x + 4) = x^2 + 4x - 2x - 8 = x^2 + 2x - 8$$ 7. **Combine the fractions:** $$\frac{x^2 + 2x - 8 + x + 10}{(x + 2)(x + 4)} = \frac{x^2 + 3x + 2}{(x + 2)(x + 4)}$$ 8. **Factor the numerator:** $$x^2 + 3x + 2 = (x + 1)(x + 2)$$ 9. **Simplify the fraction by canceling common factors:** $$\frac{\cancel{(x + 2)}(x + 1)}{\cancel{(x + 2)}(x + 4)} = \frac{x + 1}{x + 4}$$ **Final answer:** $$\frac{x + 1}{x + 4}$$