Let k be a field. Then Gaussian elimination over k and the Euclidean division algorithm for the univariate polynomial ring k[x] allow us to write any matrix in SL_n(k) or SL_n(k[x]), n\geq 2, as a product of elementary matrices. Suslin's stability theorem states that the same is true for SL_n(k[x_1,\ldots ,x_m]) with n\geq 3 and m\geq 1. In this paper, we present an algorithmic proof of Suslin's stability theorem, thus providing a method for finding an explicit factorization of a given polynomial matrix into elementary matrices. Groebner basis techniques may be used in the implementation of the algorithm.