I've done:
p[x_] := x^2 + x + 1;
roots = Solve[p[x] == 0, x] // ComplexExpand;
roots = roots /. x -> y;
Times @@ (x - y /. roots)
Which produces this output:
(* (1/2 - (I Sqrt[3])/2 + x) (1/2 + (I Sqrt[3])/2 + x) *)
What can I add so that the output is:
$$\left(x+\frac12+\frac{i\sqrt3}{2}\right)\left(x+\frac12-\frac{i\sqrt3}{2}\right)$$
I did try TraditionalForm.