HMMT 2013 A4

Determine all real values of $A$ for which there exist distinct complex numbers $x_{1}, x_{2}$ such that the following three equations hold:

$$\begin{aligned} x_{1}\left(x_{1}+1\right) & =A \\ x_{2}\left(x_{2}+1\right) & =A \\ x_{1}^{4}+3 x_{1}^{3}+5 x_{1} & =x_{2}^{4}+3 x_{2}^{3}+5 x_{2} . \end{aligned}$$