1. Stating the problem: Simplify the expression $-\frac{1}{10} + \frac{5}{8}$.\n\n2. To add or subtract fractions, they must have a common denominator. The denominators here are 10 and 8.\n\n3. Find the least common denominator (LCD) of 10 and 8.\n$$\text{LCD}(10,8) = 40$$\n\n4. Convert each fraction to an equivalent fraction with denominator 40:\n$$-\frac{1}{10} = -\frac{1 \times 4}{10 \times 4} = -\frac{4}{40}$$\n$$\frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40}$$\n\n5. Now add the fractions:\n$$-\frac{4}{40} + \frac{25}{40} = \frac{-4 + 25}{40} = \frac{21}{40}$$\n\n6. The fraction $\frac{21}{40}$ cannot be simplified further because 21 and 40 have no common factors other than 1.\n\nFinal answer: $$\frac{21}{40}$$