RMO Practice Problem - Algebra 2

Algebra Level 5

Let $(a,b)$ be the pair of integers such that

$\large ax^{17} + bx^{16} +1$

is divisible by $x^2 - x -1$ .

Then find the sum of all such pairs.

Details and Assumptions:

If you get $(2,3)$ and $(3,4)$ as your answer, then express your answer as sum of all those values i.e. $2+3+3+4=12$ .