  In this paper we show how to derive the Bogomolny's equations of the
generalized self-dual Maxwell-Chern-Simons-Higgs model presented in
\cite{Bazeia:2012ux} by using the BPS Lagrangian method with a particular
choice of the BPS Lagrangian density. We also show that the identification,
potential terms, and Gauss's law constraint can be derived rigorously under the
BPS Lagrangian method. In this method, we find that the potential terms are the
most general form that could have the BPS vortex solutions. The Gauss's law
constraint turns out to be the Euler-Lagrange equations of the BPS Lagrangian
density. We also find another BPS vortex solutions by taking other
identification between the neutral scalar field and the electric scalar
potential field, $N=\pm A_0$, which is different by a relative sign to the
identification in \cite{Bazeia:2012ux}, $N=\mp A_0$. Under this identification,
$N=\pm A_0$, we obtain a slightly different potential terms and Bogomolny's
equations compared to the ones in \cite{Bazeia:2012ux}. Furthermore we compute
the solutions numerically, with the same configurations as in
\cite{Bazeia:2012ux}, and find that only the resulting electric field plots
differ by sign relative to the results in \cite{Bazeia:2012ux}. Therefore we
conclude that these BPS vortices are electric-dual BPS vortices of the ones
computed in \cite{Bazeia:2012ux}.
