If $a^2+b^2+c^2+ab+bc+ca \le 0\quad \forall a, b, c\in\mathbb{R}$,
then find the value of the determinant
$$ \begin{vmatrix} (a+b+2)^2 & a^2+b^2 & 1 \\ 1 & (b+c+2)^2 & b^2+c^2 \\ c^2+a^2 & 1 & (c+a+2)^2 \\ \end{vmatrix}$$
I tried expanding the whole squares and using the identity $(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$, but the result is incorrect. How should I evaluate this ?
Hint $$\dfrac{1}{2}[(a+b)^2+(b+c)^2+(a+c)^2]=a^2+b^2+c^2+ab+ac+bc\le 0$$ so we have $$a=-b,b=-c,c=-a\Longrightarrow a=b=c=0$$