Can someone tell me how I can tell that sqrrt(x+4 = -5 has no solution without solving it?
Ok, I'm not sure, so if I'm wrong, someone please correct me.
We use $\displaystyle \sqrt{x}$ to represent the principal, positive square root of a number. The negative root is represented by $\displaystyle - \sqrt{x}$. The two square roots are represented together by $\displaystyle \pm \sqrt{x}$.
$\displaystyle \sqrt{x+4} = -5$ has no solution because you are referring to the positive root, which obviously can't be -5.