Hm, I don't think it is what you've been asked.
If there is a root alpha of g, then any can be transformed in .
This means that any element of E won't contain any (or highest powers).
Actually, the elements of E will be all polynomials of degree < 3, which is the degree of g(x).
So it's 0, 1, x, x+1, x², x²+x, x²+1, x²+x+1.
The transition x <-> can be done because is an equivalent class of x (sort of).
I don't know if it is clear
As I told you in another thread, this is like working in R[X]/(g(x)), which is actually the set of all possible remainders of the division of any polynomial by g(x)...
Perhaps Isomorphism will be able to explain it better