${{x}^{2}}+2{{y}^{2}}+2xy-2x+2=0$
$\Leftrightarrow 2{{x}^{2}}+4{{y}^{2}}+4xy-4x+4=0$
$\Leftrightarrow ({{x}^{2}}+4{{y}^{2}}+4xy)+\left( {{x}^{2}}-4x+4 \right)=0$
$\Leftrightarrow {{\left( x+2y \right)}^{2}}+{{\left( x-2 \right)}^{2}}=0$
$\Leftrightarrow \left\{ \begin{align}& {{\left( x+2y \right)}^{2}}=0 \\ & {{\left( x-2 \right)}^{2}}=0 \\ \end{align} \right.$
$\Leftrightarrow \left\{ \begin{align}& x+2y=0 \\ & x-2=0 \\ \end{align} \right.$
$\Leftrightarrow \left\{ \begin{align}& y=-1 \\ & x=2 \\ \end{align} \right.$