1. The problem is to subtract the mixed number $-2 \frac{1}{4}$ from $3 \frac{2}{5}$.\n\n2. First, convert the mixed numbers to improper fractions.\n\nFor $3 \frac{2}{5}$: $$3 \frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}.$$\n\nFor $-2 \frac{1}{4}$: $$-2 \frac{1}{4} = -\frac{2 \times 4 + 1}{4} = -\frac{8 + 1}{4} = -\frac{9}{4}.$$\n\n3. The subtraction becomes: $$\frac{17}{5} - \left(-\frac{9}{4}\right) = \frac{17}{5} + \frac{9}{4}.$$\n\n4. Find a common denominator to add the fractions. The least common denominator of 5 and 4 is 20.\n\nConvert each fraction: $$\frac{17}{5} = \frac{17 \times 4}{5 \times 4} = \frac{68}{20}, \quad \frac{9}{4} = \frac{9 \times 5}{4 \times 5} = \frac{45}{20}.$$\n\n5. Add the fractions: $$\frac{68}{20} + \frac{45}{20} = \frac{68 + 45}{20} = \frac{113}{20}.$$\n\n6. The result is the improper fraction $\frac{113}{20}$. To express as a mixed number: divide 113 by 20.\n\n$$113 \div 20 = 5 \text{ remainder } 13,$$ so $$\frac{113}{20} = 5 \frac{13}{20}.$$\n\nFinal answer: $$5 \frac{13}{20}.$$