prove wether the follow is a vector space or not.

could you please show me how to prove the following question? Intuituively, i think it is a vector space, but i don't know how to prove the 10 axioms step by step. Thank you very much.

Q1) S = {[x p(x)] | x is an element ofR}, Where p(x) is any polynomial, vector addition is defined as

[x p(x)] + [y p(y)] = [x+y p(x+y)]

and scalar multiplication defined as

* [x p(x)] = [ *x p( *x)]