Question 21
Determine all real values of $A$ for which there exist distinct complex numbers $x_{1}, x_{2}$ such that the following three equations hold:
$$
\begin{aligned}
x_{1}(x_{1}+1) & =A \\
x_{2}(x_{2}+1) & =A \\
x_{1}^{4}+3 x_{1}^{3}+5 x_{1} & =x_{2}^{4}+3 x_{2}^{3}+5 x_{2} .
\end{aligned}
$$
$$
\begin{aligned}
x_{1}(x_{1}+1) & =A \\
x_{2}(x_{2}+1) & =A \\
x_{1}^{4}+3 x_{1}^{3}+5 x_{1} & =x_{2}^{4}+3 x_{2}^{3}+5 x_{2} .
\end{aligned}
$$