We report on the theoretical investigation of plasmon excitations in a quasi-two-dimensional electron gas in the presence of a harmonic potential (oriented along the $x$ direction), an applied perpendicular (to the $x\text{\ensuremath{-}}y$ plane) magnetic field, and the spin-orbit interaction (SOI) induced by the Rashba effect. The resultant system is a quasi-one-dimensional (Q1D) quantum wire with free propagation along the $y$ direction and magnetoelectric quantization along the $x$. The problem involves three length scales: ${l}_{0}=\sqrt{\ensuremath{\hbar}∕{m}^{*}{\ensuremath{\omega}}_{0}}$, ${l}_{c}=\sqrt{\ensuremath{\hbar}∕{m}^{*}{\ensuremath{\omega}}_{c}}$, and ${l}_{\ensuremath{\alpha}}={\ensuremath{\hbar}}^{2}∕(2{m}^{*}\ensuremath{\alpha})$, which characterize the relative strengths in the interplay of confinement, the magnetic field, and the Rashba SOI. The resulting Schr\"odinger-like equations satisfied by the wave function (accounting for the spin-up and spin-down states) are two coupled equations, which cannot be solved in an explicit analytical form. However, invoking the limits of a strong magnetic field, ${l}_{c}⪡{l}_{0}$, and ${k}_{y}{l}_{0}⪡1$ allows us to solve this set of coupled equations exactly. We then derive and discuss the dispersion relations for charge-density excitations within the framework of Bohm-Pines' random-phase approximation. The intrasubband and intersubband magnetoplasmons in a Q1D electron gas are characterized, respectively, by the negative-energy dispersion with increasing magnetic field and the magnetoroton excitation which changes its group velocity twice before merging with the respective single-particle continuum. Here we scrutinize the effect of the Rashba SOI on these characteristics in depth. We observe that the SOI modifies drastically the behavior of both the intrasubband and intersubband magnetoplasmons in the long-wavelength limit and may render them relatively more susceptible to the Landau damping in the short-wavelength limit. We discuss the dependence of the magnetoplasmon energy on the propagation vector, the magnetic field, the 1D charge density, and the Rashba parameter characterizing the SOI.