\[\begin{array}{l}
y = {x^3} - 3\left( {m + 1} \right){x^2} + 3m\left( {m + 2} \right)x + 1\\
\Rightarrow y' = 3{x^2} - 6\left( {m + 1} \right)x + 3m\left( {m + 2} \right).\\
\Rightarrow y' = 0\\
\Leftrightarrow 3{x^2} - 6\left( {m + 1} \right)x + 3m\left( {m + 2} \right) = 0\,\\
\Leftrightarrow {x^2} - 2\left( {m + 1} \right)x + {m^2} + 2m = 0\,\,\,\,\,\left( * \right)\\
TH1:\,\,\,hs\,\,\,DB\,\,tren\,\,\,R\\
\Leftrightarrow y' \geq 0\,\,\,\forall x \in R\\
\Leftrightarrow \Delta ' \leq 0\\
\Leftrightarrow {\left( {m + 1} \right)^2} - {m^2} - 2m \leq 0\\
\Leftrightarrow {m^2} + 2m + 1 - {m^2} - 2m \leq 0\\
\Leftrightarrow 1 \leq 0\,\,\,\left( {vo\,\,ly} \right)\\
\Rightarrow hs\,\,\,k\,\,\,DB\,\,\,tren\,\,\,R.\\
TH2:\,\,pt\,\,y' = 0\,\,co\,\,2\,\,nghiem\,\,pb\\
\Leftrightarrow \Delta ' > 0 \Leftrightarrow {\left( {m + 1} \right)^2} - {m^2} - 2m > 0 \Leftrightarrow 1 > 0\,\,\,\forall m.\\
\Rightarrow pt\,\,\,\left( * \right)\,\,\,co\,\,\,2\,\,nghiem\,\,\,pb\,\,\,{x_1},\,\,{x_2}\,\,\,voi\,\,moi\,\,m.\\
\Rightarrow \left[ \begin{array}{l}
{x_1} = m + 1 + 1 = m + 2\\
{x_2} = m + 1 - 1 = m
\end{array} \right..\\
Ta\,\,co\,\,\,BXD:\\
\,\,\,\,\,\,\,\,\,\,\,\, + \,\,\,\,\,\,\,\,\,\,m\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,m + 2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \\
\Rightarrow hs\,\,\,DB\,\,\,tren\,\,\,\left( { - \infty ;\,\, - 1} \right)\,\,va\,\,\,\left( {2; + \infty } \right)\\
\Leftrightarrow \left\{ \begin{array}{l}
- 1 \le m\\
m + 2 \le 2
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
m \ge - 1\\
m \le 0
\end{array} \right. \Leftrightarrow - 1 \le m \le 0.\\
Vay\,\,\, - 1 \le m \le 0\,\,thoa\,\,man\,\,bai\,\,toan.
\end{array}\]
