no, you have to check for linear independence. P2(R) has only 3 dimensions to begin with, how could a 4 dimensional space be a subspace?
and the subspace has lots more than 4 elements, subspaces have an infinite number of elements. you're looking for a basis. a basis isn't just any collection of elements. they have to be linearly independent, and they have to span. this is important. you have to eliminate polynomials that can be written as linear combinations of other ones in your set.