But the degree of remainder must be less than that of x-1 and hence the remainders must be complex numbers. That means each equivalence class is the set of complex polynomials that leave the same complex number when divided by x-1.
However note that every complex number is clearly a remainder.And there is an equivalence class that contains it.(this shows the following map is onto)
Hence if we map each equivalence class to the complex number contained in that equivalence class, we are done!
Note: A small argument is required to establish this map is one-one