For a more "systematic" method try this. A general vector in P(3) is and your subset is defined by . We can solve for any one of the coefficients as a function of the other 3 so this is a subspace of dimension 3. For example, a= -b- c- d. Now here is a very nice method for finding bases when you have a relation like that. Take b= 1, c= d= 0 and you get a= -1. That is, -1+ x or x-1 is in the set. Take c= 1, b= d= 0. Again a= -1 so we get . Finally, take d=1, b= c= 0. Yet again a= -1 so . Taking each constant equal to 1 guarentees that those are independent so that is a basis, the same one I mentioned before.
Of course, we could have solved for any of the four coefficients. If, say, we had solved for d= -a- b- c, then: with a= 1, b=c= 0, we get . With b=1, a= c= 0, we get . With c= 1, a= b= 0, we get getting as a different basis for that subspace.