<p>This article investigates the asymptotic and finite-time synchronization of fractional-order multiplex neural networks with multiple delays. Initially, a novel extended Halanay-type inequality for fractional-order differential equations with multiple delays is developed. Based on this framework, conditions are derived to achieve asymptotic synchronization by designing adaptive control schemes. Subsequently, novel sufficient criteria for achieving finite-time synchronization are established by introducing a hybrid control protocol that incorporates the Lyapunov method, inequality techniques, and a reduction to absurdity approach. Furthermore, the settling time for synchronization is explicitly estimated. In addition, the proposed methods are extended to investigate asymptotic and finite-time synchronization for fractional-order multiplex neural networks with delay-free. In particular, the results represent a significant extension of the corresponding cases for integer-order systems. Finally, numerical simulations are provided to verify the theoretical findings. These results offer valuable insights into the synchronization of fractional-order networks with multiple delays, paving the way for scalable and practical solutions in areas such as secure communication and cross-layer integration in neural networks.</p>