Method: Solve a easier expression, then relate it to the required one.

Start with (2(n^(n-1)+(n-1)^3-(n-1)^2-1))/(n-1)^3

For n>3,

2(n^(n-1)) = 2, mod(n-1)

2(n-1)^3   = 0, mod(n-1)

2(n-1)^2   = 0, mod(n-1)

2(-1)      =-2, mod(n-1)

so the whole expression is worth 0, mod (n-1), and always an integer.

But (2(n^(n-1)+(n-1)^3-(n-1)^2-1))/(n-1)^3 = 2((n^(n-1)-1)/(n-1)^3+(n-2)/(n-1))

To approach the required expression, first divide by 2, leaving a number, (n^(n-1)-1)/(n-1)^3, plus a fraction,(n-2)/(n-1), less than 1. Even assuming that the expression remained an integer after the division by 2, once we deduct that fraction, the number, which is the required expression, is an integer diminished by 1/(n-1) and thus can never result in an integer division.