Roots of x^3 + ax^2 + bx + c = 0

[Euler_Euclid]

If $$\alpha,\beta\ and\ \gamma$$ are the roots of the eqn.

(x-a)(x-b)(x-c)+1=0

then show that $$(\alpha-x)(\beta-x)(\gamma-x)+1=0$$

 

[gunch]

Do you mean we should show that $$(\alpha-x)(\beta-x)(\gamma-x)+1=0$$ for all x? What about $$x=\alpha$$, isn't that a counterexample? Maybe I'm missing something vital, but to me it seems incorrect.