Question: Simplify.
6 3
───── + ─────
u + 1 u - 5
─────────────
5
─────
u + 1
1. **State the problem:** Simplify the expression $$\frac{\frac{6}{u+1} + \frac{3}{u-5}}{\frac{5}{u+1}}$$.
2. **Rewrite the complex fraction:** The expression is a fraction divided by another fraction. We can write it as:
$$\frac{\frac{6}{u+1} + \frac{3}{u-5}}{\frac{5}{u+1}} = \left(\frac{6}{u+1} + \frac{3}{u-5}\right) \div \frac{5}{u+1}$$
3. **Find a common denominator for the numerator:** The numerator is $$\frac{6}{u+1} + \frac{3}{u-5}$$.
The common denominator is $$(u+1)(u-5)$$.
4. **Rewrite each fraction with the common denominator:**
$$\frac{6}{u+1} = \frac{6(u-5)}{(u+1)(u-5)}$$
$$\frac{3}{u-5} = \frac{3(u+1)}{(u+1)(u-5)}$$
5. **Add the numerators:**
$$6(u-5) + 3(u+1) = 6u - 30 + 3u + 3 = 9u - 27$$
6. **So the numerator becomes:**
$$\frac{9u - 27}{(u+1)(u-5)}$$
7. **Rewrite the entire expression:**
$$\left(\frac{9u - 27}{(u+1)(u-5)}\right) \div \frac{5}{u+1} = \frac{9u - 27}{(u+1)(u-5)} \times \frac{u+1}{5}$$
8. **Cancel the common factor $u+1$:**
$$\frac{9u - 27}{\cancel{(u+1)}(u-5)} \times \frac{\cancel{u+1}}{5} = \frac{9u - 27}{5(u-5)}$$
9. **Factor numerator $9u - 27$:**
$$9u - 27 = 9(u - 3)$$
10. **Final simplified expression:**
$$\frac{9(u - 3)}{5(u - 5)}$$
**Answer:** $$\frac{9(u - 3)}{5(u - 5)}$$