$$\eqalign{
& y = {{\sin x + 2\cos x + 1} \over {\sin x + \cos x + 2}} \cr
& \Leftrightarrow \sin x + 2\cos x + 1 = y\sin x + y\cos x + 2y \cr
& \Leftrightarrow \left( {1 - y} \right)\sin x + \left( {2 - y} \right)\cos x = 2y - 1 \cr
& PT\,\,co\,\,nghiem \cr
& \Leftrightarrow {\left( {1 - y} \right)^2} + {\left( {2 - y} \right)^2} \ge {\left( {2y - 1} \right)^2} \cr
& \Leftrightarrow 1 - 2y + {y^2} + 4 - 4y + {y^2} \ge 4{y^2} - 4y + 1 \cr
& \Leftrightarrow - 2{y^2} - 2y + 4 \ge 0 \cr
& \Leftrightarrow - 2 \le y \le 1 \cr
& Vay\,\,\min y = - 2;\,\,\max y = 1 \cr} $$
