subspace test

I dont know how to prove whether a set is a subspace or not. I know the axioms (0 isin the subspace, closed under addition and scalar multiplication) But I understand what they really means when I look at some exercises. For example, x^4+ 3x^2 + 2,
W= {(x,y,z) | x-y+z=0, x+y=0 } ....
How to I know initially whether it is closed under addition/ multiplication or not?


Can anyone explain further the concept and give some examples and explain why
they are subspaces?
also, if anyone knows any website that provides more exerecise and examples of
subspaces, plz type here too. Thx.
(I did research on the Internet before I post this question but I still dont understand)