To divide \(-\frac{10}{11}\) by \(-\frac{2}{5}\), you can multiply by the reciprocal of the second fraction:
\[ -\frac{10}{11} \div -\frac{2}{5} = -\frac{10}{11} \times -\frac{5}{2} \]
Now, multiplying the fractions:
\[ \frac{10 \times 5}{11 \times 2} = \frac{50}{22} \]
Next, simplify \(\frac{50}{22}\):
\[ \frac{50}{22} = \frac{25}{11} \]
Now we can convert \(\frac{25}{11}\) to a mixed number:
\[ 25 \div 11 = 2 \quad \text{(remainder 3)} \]
So, \(\frac{25}{11} = 2 \frac{3}{11}\).
Therefore, the quotient when you divide \(-\frac{10}{11}\) by \(-\frac{2}{5}\) is \(\boxed{2 \frac{3}{11}}\).