<p>In this work, Asymptotic Iteration Method (AIM) is systematically employed to obtain quasi-exact analytical solutions of the Schrödinger equation for a one-dimensional general sextic oscillator which also includes a centrifugal term. This potential function has been previously discussed in the literature and quasi-exact eigenfunctions and eigenvalues of the corresponding quantum mechanical system have been investigated by means of the bi-confluent Heun equation and quasi-exact solvability approach. Alternatively, we achieve the energy eigenvalues in a closed form, which can be used to establish energy spectrum, for the case where none of potential parameters equals zero, in the present study. For reliability analysis of the analytical expression obtained, a numerical scheme based on AIM is further utilized within the arbitrary potential parameter regimes. Owing to this analytical expression, one will be able to see the effect of each potential parameters on the energy spectrum of the system.</p>