This article is a proposal for the teaching and learning of factoring quadratic polynomials, based Regine Douady's theoretical frameworks regarding the Tool-Object Dialectic. It highlights the role of combined mathematical frameworks as a means to access... "different formulations of a problem which, without necessarily being equivalent, allow a new approach to the difficulties encountered and the staging of tools and techniques that were not imposed in the first formulation." From this perspective, an axiomatic system is constructed in a figural context in which primary objects are defined, and relationships are established between them, leading to definitions and theorems of a mathematical theory within this framework. Rules are proposed for the operation of the primary objects and a game is described in which the factoring is operationalized using this framework. The game becomes the vehicle for understanding the difficult relations of the objects of factoring in the algebraic framework. This form of educational work develops cognitive and metacognitive skills in students who discover and describe the factorization properties, establish the algebraic relations between the coefficients of the polynomial and the factors that factored expression, developing a mathematical activity.