<p>The presented study investigates exact results to the high-dispersion cubic-quintic nonlinear Schrödinger equation (HD-CQNLSE), which was chosen because it provides a more realistic description of nonlinear wave propagation in optical media compared to the standard NLS model. Specifically, it incorporates high-order dispersion and cubic-quantic nonlinearity, which are essential elements for modeling ultrashort optical pulses, high-intensity wave propagation, and non-kerr media. The modified extended direct algebric method (MEDAM) was chosen not only because of its relatively limited use, but also because of its flexibility and ability to handle higher-order nonlinear equations. Unlike more restrictive analytical methods, MEDAM allows for a wider range of solution structures through its algebraic formulation, including hyperbolic (solitary wave) solutions, trigonometric (periodic) solutions, and singular solutions. These solutions are important for providing an analytical understanding of the role of higher-order scattering and nonlinear effects in wave propagation. Furthermore, we present three-dimensional graphical representations and two-dimensional drawings illustrating the effect of changing the equation parameters on the waveform, which, along with the extracted solutions, enrich the analytical understanding of the behavior of nonlinear wave dynamics under high dispersion and lay the foundation for potential applications in nonlinear optics and related fields. Finally, we apply linear stability analysis (LSA) to HD-CQNLSE. These results are important for understanding wave collapse, soliton formation, and rogue wave dynamics in optical fibers, Bose-Einstein condensates (BECs), and other nonlinear wave systems.</p>