Abstract
Sextic polynomial oscillator is probably the best known quantum system which is partially exactly {\it alias} quasi-exactly solvable (QES), i.e., which possesses closed-form, elementary-function bound states ψ(x) at certain couplings and energies. In contrast, the apparently simpler and phenomenologically more important quartic polynomial oscillator is {\em not\,} QES. A resolution of the paradox is proposed: The one-dimensional Schr\"{o}dinger equation is shown QES after the analyticity-violating symmetrization V(x)=A|x|+B x²+C|x|³+x⁴ of the quartic polynomial potential.