${{z}^{2}}+2z+5=0\Leftrightarrow \left[ \begin{align}  & {{z}_{1}}=-1+2i \\  & {{z}_{2}}=-1-2i \\ \end{align} \right.$$\Rightarrow \left[ \begin{align}  & \overline{{{z}_{1}}}=-1-2i \\  & \overline{{{z}_{2}}}=-1+2i \\ \end{align} \right.$

$\begin{align}  & \left| {{z}_{1}} \right|=\sqrt{1+4}=\sqrt{5}\text{  ;  }\left| {{z}_{2}} \right|=\sqrt{1+4}=\sqrt{5} \\  & \left| \overline{{{z}_{1}}} \right|=\sqrt{1+4}=\sqrt{5}\text{  ; }\left| \overline{{{z}_{2}}} \right|=\sqrt{1+4}=\sqrt{5} \\ \end{align}$

Vậy  $A=5+5-4.\sqrt{5}.\sqrt{5}=-10$.