Apart[(1/(1-x-(x^2)))]

Factor[(1/(1-x-(x^2)))]

Factor[(1/(1-x-(x^2))),Extension->Root[1-x-(x^2),1]]

Factor[(1/(1-x-(x^2))),Extension->Root[1-x-(x^2),1]] // Apart

(Factor[(1/(1-x-(x^2))),Extension->Root[1-x-(x^2),1]] // Apart) /. ((a_)/(b_+c_*x)):>(a/b)(((-(c/b)))^n)

(Factor[(1/(1-x-(x^2))),Extension->Root[1-x-(x^2),1]] // Apart) /. ((a_)/(b_+c_*x)):>(a/b)(((-(c/b)))^n) /. n->100 // Simplify