Question 42
Given that $a$, $b$, and $c$ are complex numbers satisfying
$$
\begin{aligned}
a^{2}+a b+b^{2} & =1+i \\
b^{2}+b c+c^{2} & =-2 \\
c^{2}+c a+a^{2} & =1,
\end{aligned}
$$
compute $(a b+b c+c a)^{2}$. (Here, $i=\sqrt{-1}$.)
$$
\begin{aligned}
a^{2}+a b+b^{2} & =1+i \\
b^{2}+b c+c^{2} & =-2 \\
c^{2}+c a+a^{2} & =1,
\end{aligned}
$$
compute $(a b+b c+c a)^{2}$. (Here, $i=\sqrt{-1}$.)