Any "polynomial of degree at most 5" can be written as . f(0)= f(1) means that . f(0)= f(-1) means that .Find a basis for the space of all polynomials of degree at most 5 for which .
With those two equations you should be able to write two of the coefficients in terms of the other four. Now choose one of those four to be 1, the others 0, etc.