1. The problem is to divide the fraction $\frac{-2}{9}$ by $-4$. 2. Recall the division of fractions rule: dividing by a number is the same as multiplying by its reciprocal. 3. So, $\frac{-2}{9} \div (-4) = \frac{-2}{9} \times \frac{1}{-4}$. 4. Multiply the numerators and denominators: $$\frac{-2 \times 1}{9 \times -4} = \frac{-2}{-36}$$ 5. Simplify the fraction by canceling the negatives: $$\frac{\cancel{-2}}{\cancel{-36}} = \frac{2}{36}$$ 6. Simplify $\frac{2}{36}$ by dividing numerator and denominator by 2: $$\frac{\cancel{2}}{\cancel{36}} = \frac{1}{18}$$ 7. Final answer: $\frac{1}{18}$.