  We develop an approach to the deformation quantization on the real plane with
an arbitrary Poisson structure which based on Weyl symmetrically ordered
operator products. By using a polydifferential representation for deformed
coordinates $\hat x^j$ we are able to formulate a simple and effective
iterative procedure which allowed us to calculate the fourth order star product
(and may be extended to the fifth order at the expense of tedious but otherwise
straightforward calculations). Modulo some cohomology issues which we do not
consider here, the method gives an explicit and physics-friendly description of
the star products.
