1. **State the problem:** Simplify the expression $$2 \frac{5}{8} - \left(-\frac{1}{5}\right) \div \left(-\frac{8}{5}\right) + \frac{1}{4}$$. 2. **Convert mixed number to improper fraction:** $$2 \frac{5}{8} = \frac{2 \times 8 + 5}{8} = \frac{21}{8}$$. 3. **Rewrite the expression:** $$\frac{21}{8} - \left(-\frac{1}{5}\right) \div \left(-\frac{8}{5}\right) + \frac{1}{4}$$. 4. **Divide the fractions:** Dividing by a fraction is multiplying by its reciprocal: $$-\frac{1}{5} \div -\frac{8}{5} = -\frac{1}{5} \times -\frac{5}{8}$$. 5. **Multiply the fractions:** $$-\frac{1}{5} \times -\frac{5}{8} = \frac{1 \times 5}{5 \times 8} = \frac{5}{40}$$. 6. **Simplify the fraction:** $$\frac{5}{40} = \frac{\cancel{5}^1}{\cancel{5}^8} = \frac{1}{8}$$. 7. **Substitute back:** $$\frac{21}{8} - \frac{1}{8} + \frac{1}{4}$$. 8. **Find common denominator (8):** $$\frac{21}{8} - \frac{1}{8} + \frac{1}{4} = \frac{21}{8} - \frac{1}{8} + \frac{2}{8}$$. 9. **Perform addition and subtraction:** $$\frac{21 - 1 + 2}{8} = \frac{22}{8}$$. 10. **Simplify the fraction:** $$\frac{22}{8} = \frac{\cancel{2}11}{\cancel{2}4} = \frac{11}{4}$$. **Final answer:** $$\frac{11}{4}$$ or as a mixed number $$2 \frac{3}{4}$$.