  We give a unified method to derive the strong convergence rate of the
backward Euler scheme for monotone SDEs in $L^p(\Omega)$-norm, with general $p
\ge 4$. The results are applied to the backward Euler scheme of SODEs with
polynomial growth coefficients. We also generalize the argument to the
Galerkin-based backward Euler scheme of SPDEs with polynomial growth
coefficients driven by multiplicative trace-class noise.
