How did the solution exclude one possibility?

[Loki123]

Imagine starting with
-2 = 2 (which is clearly not true).

If you square both sides, you get
4 = 4 (which is true).

That is effectively what you have done!

Then how do I check the solution?

 

[Harry_the_cat]

Substitute your solution/s into the LHS and RHS seperately.
Or, just be aware that the RHS cannot be negative, so you can toss out one solution straight away.