1. **State the problem:** Calculate the sum of $-4 \frac{3}{8}$ and $6 \frac{1}{2}$.\n\n2. **Convert mixed numbers to improper fractions:**\n$-4 \frac{3}{8} = -\left(4 + \frac{3}{8}\right) = -\frac{32}{8} - \frac{3}{8} = -\frac{35}{8}$\n$6 \frac{1}{2} = 6 + \frac{1}{2} = \frac{12}{2} + \frac{1}{2} = \frac{13}{2}$\n\n3. **Find a common denominator:**\nThe denominators are 8 and 2. The least common denominator (LCD) is 8.\nConvert $\frac{13}{2}$ to eighths: $\frac{13}{2} = \frac{13 \times 4}{2 \times 4} = \frac{52}{8}$\n\n4. **Add the fractions:**\n$-\frac{35}{8} + \frac{52}{8} = \frac{-35 + 52}{8} = \frac{17}{8}$\n\n5. **Convert the improper fraction back to a mixed number:**\n$\frac{17}{8} = 2 \frac{1}{8}$\n\n**Final answer:** $2 \frac{1}{8}$