Question 44804
x^4+x^3+x+1 

If you substitute x=-1 you get zero.
So -1 is a solution.
Divide by x+1 to get x^3+1
This also has a zero of -1
The remaining factor is x^2-x+1
Use the quadratic formula if you want but
you won't find any other integer solutions.

So the interger solutions are -1 with multiplicity two.

Cheers,
Stan H.