  For the displaced harmonic double-well oscillator the existence of exact
polynomial bound states at certain displacements $d\,$ is revealed. The
$N-$plets of these quasi-exactly solvable (QES) states are constructed in
closed form. For non-QES states, Schr\"{o}dinger equation can still be
considered ``non-polynomially exactly solvable'' (NES) because the exact left
and right parts of the wave function (proportional to confluent hypergeometric
function) just have to be matched in the origin.
