\[\begin{array}{l} y = {x^3} - 3\left( {2m + 1} \right){x^2} + \left( {12m + 5} \right)x + 2\\ \Rightarrow y' = 3{x^2} - 6\left( {2m + 1} \right)x + 12m + 5\\ \Rightarrow y' = 0\\ \Leftrightarrow 3{x^2} - 6\left( {2m + 1} \right)x + 12m + 5 = 0\\ \Delta ' = 9{\left( {2m + 1} \right)^2} - 3\left( {12m + 5} \right) = 36{m^2} + 36m + 9 - 36m - 15\\ = 36{m^2} - 6.\\ TH1:\,\,\,Hs\,\,DB\,\,\,tren\,\,\,R\\ \Leftrightarrow \Delta ' < 0 \Leftrightarrow 36{m^2} - 6 < 0 \Leftrightarrow {m^2} < \frac{1}{6} \Leftrightarrow - \frac{1}{{\sqrt 6 }} < m < \frac{1}{{\sqrt 6 }}\\ TH2:\,\,\,Pt\,\,y' = 0\,\,\,co\,\,\,2\,\,\,nghiem\,\,\,pb\\ \Leftrightarrow \Delta ' > 0 \Leftrightarrow \left[ \begin{array}{l} m > \frac{1}{{\sqrt 6 }}\\ m < - \frac{1}{{\sqrt 6 }} \end{array} \right..\\ Bang\,\,xet\,\,dau:\\ \,\,\,\,\,\,\,\,\,\,\,\,\, + \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x_1}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x_2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \\ Hs\,\,\,DB\,\,tren\,\,\,\left( {2; + \infty } \right)\\ \Leftrightarrow {x_1} < {x_2} \le 2\\ \Leftrightarrow \left\{ \begin{array}{l} {x_1} + {x_2} < 4\\ \left( {{x_1} - 2} \right)\left( {{x_2} - 2} \right) \ge 0 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} 2\left( {2m + 1} \right) < 4\\ {x_1}{x_2} - 2\left( {{x_1} + {x_2}} \right) - 4 \ge 0 \end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l} 2m + 1 < 2\\ \frac{{12m + 5}}{3} - 2.2\left( {2m + 1} \right) - 4 \ge 0 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} 2m < 1\\ 12m + 5 - 24m - 12 - 12 \ge 0 \end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l} m < \frac{1}{2}\\ - 12m \ge - 19 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} m < \frac{1}{2}\\ m \le \frac{{19}}{{12}} \end{array} \right. \Leftrightarrow m < \frac{1}{2}\\ Ket\,\,hop\,\,voi\,\,dieu\,\,kien\,\,\left[ \begin{array}{l} m > \frac{1}{{\sqrt 6 }}\\ m < - \frac{1}{{\sqrt 6 }} \end{array} \right. \Rightarrow \left[ \begin{array}{l} \frac{1}{{\sqrt 6 }} < m < \frac{1}{2}\\ m < - \frac{1}{{\sqrt 6 }} \end{array} \right..\\ Vay\,\,\,\,m < \frac{1}{2};\,\,m \ne \frac{1}{{\sqrt 6 }}\,\,\,thoa\,\,\,man\,\,bai\,\,toan. \end{array}\]