A basis is just a collect of vectors that describe the whole space you are dealing with. In geometry we usually write any three dimensional vector as V = ai + bj + ck where a,b,c are real numbers and i,j,k are vectors corresponding to the x,y,z axis in 3D geometry. They don't have to be these specific vectors but this is a common basis: as long you can construct any vector in the space using a linear combination of the basis vectors, then the basis vectors constitute a basis.
Instead of having a vector in <x,y,z> you have a polynomial in <x^2,x,1> where you have basis vectors x,y,z such that X = ax + by + cz describes all possible vectors X in your space.
Now you need to identify a basis for the polynomial and the first step is to think of the x^2 terms independent to x terms and x terms independent to constant terms: in fact all three will be independent from each other and then think of these as vectors <x,y,z> instead of <x^2,x,1>