  The contributions from the color-octet components $|(c\bar b)_{\bf
8}(^{1}S_{0}) g>$ and $|(c\bar b)_{\bf 8}(^{3}S_{1}) g>$ to the $h_{B_c}$ or
$\chi_{B_c}^J$ (the $P$-wave $B_c$ meson) hadroproduction are estimated in
terms of the complete ${\cal O}(\alpha_s^4)$ calculation. As necessary inputs
in the estimate, we take the values of the octet matrix elements according to
the NRQCD scaling rules, and as a result, we have found that the contributions
to the $P$-wave production may be the same in order of magnitude as those from
the color-singlet ones, $|(c\bar b)_{\bf 1}(^{1}P_{1})>$ and $|(c\bar b)_{\bf
1}(^{3}P_{J})>$ ($J=1,2,3$). Especially, our result indicates that the
observation of the color-octet contributions to the $P$-wave production in the
low transverse momentum region is not very far from the present experimental
capability at Tevatron and LHC.
