Call $ x_1,x_2,x_3$ is three integer root of the polynomial. Apply viete law we has: $ x_1+x_2+x_3=0$ $ x_1x_2+x_2x_3+x_3x_1=-1$ Imply that : $ x_1x_2-(x_1+x_2)^2=-1$ $ \Leftrightarrow x_1^2+x_1x_2+x_1^2=1$ $ \Rightarrow x_1x_2x_3=0$ $ \Rightarrow a=0$ With $ a=0$ then the equation has solution $ x\in\{-1,1,0\}$