\[\frac{a^{2}b \cdot \left( ab^{2} \right)^{2}}{a^{3}b^{4}} = \frac{a^{2}b \cdot a^{2}b^{4}}{a^{3}b^{4}} = \frac{a^{4}b^{5}}{a^{3}b^{4}} = ab\ \]