  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 $\psi(x)$ 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 $V(x)=A|x|+B x^2+C|x|^3+x^4$ of the quartic polynomial
potential.
