1. **State the problem:** Add and simplify the rational expressions $$\frac{6}{x+5} + \frac{x+65}{x^2-25}$$. 2. **Identify the denominators:** The denominators are $x+5$ and $x^2-25$. 3. **Factor the second denominator:** Note that $$x^2-25 = (x+5)(x-5)$$. 4. **Find the common denominator:** The least common denominator (LCD) is $$(x+5)(x-5)$$. 5. **Rewrite each fraction with the LCD:** - The first fraction $$\frac{6}{x+5}$$ needs to be multiplied by $$\frac{x-5}{x-5}$$: $$\frac{6}{x+5} = \frac{6(x-5)}{(x+5)(x-5)}$$ - The second fraction already has the LCD: $$\frac{x+65}{(x+5)(x-5)}$$ 6. **Add the numerators:** $$\frac{6(x-5)}{(x+5)(x-5)} + \frac{x+65}{(x+5)(x-5)} = \frac{6(x-5) + (x+65)}{(x+5)(x-5)}$$ 7. **Expand and simplify the numerator:** $$6(x-5) + (x+65) = 6x - 30 + x + 65 = 7x + 35$$ 8. **Factor the numerator:** $$7x + 35 = 7(x + 5)$$ 9. **Write the full expression:** $$\frac{7(x+5)}{(x+5)(x-5)}$$ 10. **Cancel the common factor $(x+5)$:** $$\frac{7\cancel{(x+5)}}{\cancel{(x+5)}(x-5)} = \frac{7}{x-5}$$ **Final answer:** $$\boxed{\frac{7}{x-5}}$$