\[\begin{array}{l}\left| {2{x^2} + {y^2} - 1} \right| + {y^2} + 4y = 0 \Leftrightarrow \\\left\{ \begin{array}{l}{y^2} + 4y \le 0\\\left[ \begin{array}{l}2{x^2} + {y^2} - 1 + {y^2} + 4y = 0\\ - 2{x^2} - {y^2} + 1 + {y^2} + 4y = 0\end{array} \right.\end{array} \right. \Leftrightarrow \\\left\{ \begin{array}{l}y\left( {y + 4} \right) \le 0\\\left[ \begin{array}{l}{x^2} + {\left( {y + 1} \right)^2} = \frac{3}{2}\\y = \frac{1}{4}\left( {2{x^2} - 1} \right)\end{array} \right.\end{array} \right.\end{array}\]