Summary: \textit{Milliken-Taylor systems} are some of the most general infinitary configurations that are known to be partition regular. These are sets of the form \(\mathrm{MT}(\langle a_i\rangle _{i=1}^m,\langle x_n\rangle _{n=1}^\infty )= \{\sum _{i=1}^m a_i\sum _{t\in F_i}\,x_t:F_1,F_2,\ldots , F_m\) are increasing finite nonempty subsets of \( \mathbb{N}\}\), where \( a_1,a_2,\ldots ,a_m\in \mathbb{Z}\) with \( a_m>0\) and \( \langle x_n\rangle _{n=1}^\infty \) is a sequence in \( \mathbb{N}\). That is, if \( p(y_1,y_2,\ldots ,y_m)=\sum _{i=1}^m a_iy_i\) is a given linear polynomial and a finite coloring of \( \mathbb{N}\) is given, one gets a sequence \( \langle x_n\rangle _{n=1}^\infty \) such that all sums of the form \( p(\sum _{t\in F_1}x_t,\ldots ,\sum _{t\in F_m}x_t)\) are monochromatic. In this paper we extend these systems to images of very general \textit{extended polynomials}. We work with the Stone-Čech compactification \( \beta {\mathcal F}\) of the discrete space \( {\mathcal F}\) of finite subsets of \( \mathbb{N}\), whose points we take to be the ultrafilters on \( {\mathcal F}\). We utilize a simply stated result about the tensor products of ultrafilters and the algebraic structure of \( \beta {\mathcal F}\).