$\displaystyle \begin{array}{{>{\displaystyle}l}} x^{2} \ -\ 2x+\ 2-m=0\ ( 1)\\ \Delta '=1-2+m=m-1\\ ( 1) \ có\ 2\ nghiệm\ \Leftrightarrow m-1\geqslant 0\ \Leftrightarrow m\geqslant 1\\ Theo\ Viet:\ x_{1} +x_{2} =2\ \ \ \ \ x_{1} x_{2} =2-m\\ \Rightarrow x_{2} =2-x_{1} ;\ m=2-x_{1} x_{2} =2-x_{1}( 2-x_{1}) =x^{2}_{1} -2x_{1} +2\\ 2x^{3}_{1} +( m+2) x^{2}_{2} =5\\ \Leftrightarrow 2x^{3}_{1} +\left( \ x^{2}_{1} -2x_{1} +4\right)\left( x^{2}_{1} +4-4x_{1}\right) =5\\ \Leftrightarrow x_{1} =1\Rightarrow x_{2} =1\\ \Rightarrow m=2-1.1=1\ ( tm)\\ \end{array}$