[tex]S = \Big\{x\in \mathbb{R}\,\Big|\,\,x^2-(m+2)x+m+1 = 0\Big\} = \{1\} \\ \\S = \{1\}\\ \\ \Rightarrow x^2-(m+2)x+m+1 = 0 \quad\to\quad x \in \{1\}\text{ (solutie unica)} \\\\ \\ \Rightarrow \begin{cases}\Delta = 0\\ x^2-(m+2)x+m+1\Big|_{x=1} = 0\end{cases} \\ \\\\ \Rightarrow \begin{cases}(m+2)^2-4(m+1) = 0 \\ 1-(m+2)\cdot 1+m+1 = 0\end{cases} \\ \\\\ \Rightarrow \begin{cases}m^2+4m+4-4m-4=0\\1-m-2+m+1 = 0\end{cases}[/tex]
[tex]\\\Rightarrow \begin{cases}m^2 = 0 \\ 0\cdot m = 0 \end{cases} \Rightarrow \begin{cases}m= 0 \\ m\in \mathbb{R} \end{cases}\Bigg|\Rightarrow m\in \{0\}\cap \mathbb{R}\,\Rightarrow\,\boxed{m = 0}[/tex]
