  Recently, Lembert, Gilson et al proposed a lucid and systematic approach to
obtain bilinear B\"{a}cklund transformations and Lax pairs for
constant-coefficient soliton equations based on the use of binary Bell
polynomials. In this paper, we would like to further develop this method with
new applications. We extend this method to systematically investigate complete
integrability of nonisospectral and variable-coefficient equations. In
addiction, a method is described for deriving infinite conservation laws of
nonlinear evolution equations based on the use of binary Bell polynomials. All
conserved density and flux are given by explicit recursion formulas. By taking
variable-coefficient KdV and KP equations as illustrative examples, their
bilinear formulism, bilinear B\"{a}cklund transformations, Lax pairs, Darboux
covariant Lax pairs and conservation laws are obtained in a quick and natural
manner. In conclusion, though the coefficient functions have influences on a
variable-coefficient nonlinear equation, under certain constrains the equation
turn out to be also completely integrable, which leads us to a canonical
interpretation of their $N$-soliton solutions in theory.
