1. **State the problem:** We need to add the two fractions $$\frac{5}{n+5} + \frac{4n}{2n+6}$$. 2. **Identify the denominators:** The denominators are $n+5$ and $2n+6$. 3. **Factor the second denominator:** $$2n+6 = 2(n+3)$$ 4. **Find the least common denominator (LCD):** The LCD must include both $n+5$ and $2(n+3)$, so it is: $$2(n+5)(n+3)$$ 5. **Rewrite each fraction with the LCD:** For the first fraction: $$\frac{5}{n+5} = \frac{5 \times 2(n+3)}{2(n+5)(n+3)} = \frac{10(n+3)}{2(n+5)(n+3)}$$ For the second fraction: $$\frac{4n}{2(n+3)} = \frac{4n \times (n+5)}{2(n+5)(n+3)} = \frac{4n(n+5)}{2(n+5)(n+3)}$$ 6. **Add the numerators:** $$10(n+3) + 4n(n+5)$$ 7. **Expand the terms:** $$10n + 30 + 4n^2 + 20n$$ 8. **Combine like terms:** $$4n^2 + (10n + 20n) + 30 = 4n^2 + 30n + 30$$ 9. **Write the final expression:** $$\frac{4n^2 + 30n + 30}{2(n+5)(n+3)}$$ 10. **Factor numerator if possible:** Factor out 2: $$2(2n^2 + 15n + 15)$$ No simple factorization for $2n^2 + 15n + 15$. 11. **Simplify the fraction by canceling 2:** $$\frac{\cancel{2}(2n^2 + 15n + 15)}{\cancel{2}(n+5)(n+3)} = \frac{2n^2 + 15n + 15}{(n+5)(n+3)}$$ **Final answer:** $$\frac{2n^2 + 15n + 15}{(n+5)(n+3)}$$