1. Stating the problem: Divide the mixed fraction $-\frac{2}{5}_3$ by the mixed number $3\frac{6}{7}$. 2. First, convert the mixed numbers to improper fractions. For $-\frac{2}{5}_3$, assuming it means $-3\frac{2}{5}$, convert as follows: $$-3\frac{2}{5} = -\left(3 + \frac{2}{5}\right) = -\frac{15}{5} - \frac{2}{5} = -\frac{17}{5}$$ 3. Convert $3\frac{6}{7}$ to an improper fraction: $$3\frac{6}{7} = 3 + \frac{6}{7} = \frac{21}{7} + \frac{6}{7} = \frac{27}{7}$$ 4. Division of fractions is multiplication by the reciprocal: $$-\frac{17}{5} \div \frac{27}{7} = -\frac{17}{5} \times \frac{7}{27}$$ 5. Multiply numerators and denominators: $$-\frac{17 \times 7}{5 \times 27} = -\frac{119}{135}$$ 6. Simplify the fraction if possible. The greatest common divisor of 119 and 135 is 1, so the fraction is already in simplest form. Final answer: $$-\frac{119}{135}$$