Question 1036408
The remainder would be {{{x^2 - x - 1}}}.

To see this, notice that the polynomial {{{f(x) = x^(2n-1) - x - 1}}}, where {{{n>=2}}}, will always give a remainder of -1 upon division by x-1 and a remainder of -x-1 upon division by {{{x^2}}}.  By using synthetic division with x - 1 as divisor, it can be easily seen that f(x) is unique in form.  

By applying the usual polynomial division, the remainder after dividing {{{f(x) = x^(2n-1) - x - 1}}} by {{{x^2(x-1) = x^3 - x^2}}} is {{{x^2 - x - 1}}}.