Accurate trajectory prediction is crucial for safe and reliable autonomous driving systems, requiring models that capture long-term temporal dependencies while accounting for social interactions among neighboring vehicles in highway driving scenarios. While Long Short Term Memory (LSTM) networks have been widely used in the domain of trajectory prediction, they have limitations such as limited memory capacity and scalar cell state. The recently introduced Extended Long Short Term Memory (xLSTM) addresses these limitations of traditional LSTMs by introducing exponential gating and enhanced memory structures, making them better suited for modeling long-term temporal dependencies. Despite their potential, xLSTM-based models remain underexplored in the context of vehicle trajectory prediction. This paper introduces a novel xLSTM-based highway trajectory prediction framework, X-TRAJ, as the first application of xLSTM, and its physics-aware variant, X-TRACK (eXtended LSTM for TRAjectory prediction Constraint by Kinematics), which explicitly integrates vehicle motion kinematics into the model learning process. By introducing physical constraints, the proposed model generates realistic and feasible highway trajectories. A comprehensive evaluation on the publicly available highway datasets, highD and NGSIM, demonstrates that X-TRACK outperforms state-of-the-art baselines on highD and is among the state-of-the-art models on the NGSIM dataset.