thevinh, you are absoluteley right. You can lose info by taking square root.
(x+2)^2<1 is unambiguous.
-3<x<-1, which is what you did.
However, back to OP. Plato is right, taking square roots can lead to ambiguities (and loss of solutions as thevinh points out)
1) a^2>0 -> a='0, or
2) a^2>0 -> a>0, or
3) a^2>0 -> lal>0 -> a>0 or a<0.
The only thing pointing to 2) or 3) is that they lead to quadratic inequalities, the title of the OP.