Abstract: This article presents a detailed analytical approach to solving the fractional reaction-subdiffusion equation, a vital model in understanding anomalous diffusion processes. The study introduces a method that combines the Laplace transform with the Adomian decomposition method to derive solutions. This approach is significant as it addresses the complexity inherent in fractional differential equations which are used to model natural phenomena with memory effects. Key Methodology: By using the Laplace transform, the authors convert the original fractional reaction-subdiffusion equation into a more manageable form. Then, the Adomian decomposition method is applied to obtain series solutions. This hybrid technique is demonstrated to be effective in solving fractional equations without requiring linearization or discretization, thus maintaining the integrity of the original problem. Results: The solutions derived are validated against known special cases, confirming the accuracy and efficiency of the proposed method. The methodology is versatile and applicable to a broad class of fractional differential equations beyond the specific model studied. Conclusion: This work advances the analytical techniques available for fractional calculus, offering a robust tool for researchers dealing with complex diffusion models in various scientific fields.