Using the factor theorem the roots multiplied together have to be -1 or 1 (the constant term in the equation)
This is a small proof od the factor theorem.
The constant on the left hand side -pqr is equal to the constant on the right hand side -1The cubic equation should have three roots, call these roots p, q and r
When these are the three roots
(a-p)(a-q)(a-r)= a^3 + a^2 - a - 1
a^3 - (p+q+r)*a^2 + (pq+pr+rq)*a - pqr= a^3 + a^2 - a - 1
So pqr= 1
Whatever the values of p, q and r are they must multiply to give 1.
It is possible that they are not integers (whole numbers) but to begin with assume that they are.
The only factors of 1 are 1 and -1
Test if a=1 is a root
(1)^3 + (1)^2 - (1) -1= 0
So 1 is a root.
a=1 is a root implies (a-1) is a factor.
use long division to divide a^3 + a^2 - a - 1 by (a-1) and you will get a quadratic equation. Solve the quadratic equation to find the other two roots.