The solution of polynomial controller design is usually reduced to certain polynomial operations. However, these operations are given in an abstract form without clear mathematical reasoning. Therefore, this paper is devoted to present a novel derivation for the problem of polynomial generalized-linear-quadratic-gaussian (GLQG) control following a systematic approach for the derivation and considering a more general plant-structure that contains colored input disturbance and measurement noise. The presentation of the theory comes in a more concise, clear and general form to help those looking to use it without any details as well as those looking for detailed understanding and tailoring the theory to their problems. The cost function includes dynamic weighting elements allowing integral action to be introduced and robustness characteristics to be modified. Thus, the novelty of the paper stems from the fact that it presents the proof in a novel approach for a general plant structure which covers any special case in reality. The paper is supplemented with design steps and two numerical examples: one is a continuous time system and the other is a discrete time system.