$\begin{array}{l} \left\{ \begin{array}{l} {c^2} + ac + b = 0;\\ {d^2} + ad + b = 0\\ {a^2} + ca + d = 0\\ {b^2} + cb + d = 0 \end{array} \right. \Rightarrow \left\{ \begin{array}{l} {c^2} - {d^2} + a\left( {c - d} \right) = 0\\ {a^2} - {b^2} + c\left( {a - b} \right) = 0 \end{array} \right.\\ \Rightarrow \left\{ \begin{array}{l} \left( {c - d} \right)\left( {c + d + a} \right) = 0\\ \left( {a - b} \right)\left( {a + b + c} \right) = 0 \end{array} \right. \Rightarrow \left\{ \begin{array}{l} c = d\\ a = b\\ a + c + d = 0\\ a + b + c = 0 \end{array} \right.\\ \Rightarrow \left\{ \begin{array}{l} c = d\\ a = b\\ a + d + d = 0\\ a + a + d = 0 \end{array} \right. \Rightarrow \left\{ \begin{array}{l} c = d\\ a = b\\ a + 2d = 0\\ 2a + d = 0 \end{array} \right.\\ \Rightarrow a = b = c = d = 0 \Rightarrow a + b + c + d = 0\\