Pasquale,

Not sure what you want to do with the roots. Here is one option if you want to show them above the plot.

Manipulate[Module[{poly = a*x^4 + b*x^2 + c*x + 2},
  Pane[roots = NSolve[poly == 0, x];
   Column[
    {
     Style[Column[Prepend[roots, "Roots"]], 10],
     Show[
      Plot[poly, {x, -20, 20}, PlotRange -> {{-20, 20}, {-20, 20}}]]
     }, Alignment -> Center]]],
 {a, -1, 1}, {b, -10, 10}, {c, -10, 10}, ControlPlacement -> Right]

You can use Solve to find the roots. e.g. with a = 1, b = 10, c = 10.

Solve[1*x^4 + 10*x^2 + 10*x + 2 == 0, x]

For a numerical approximation:

NSolve[1*x^4 + 10*x^2 + 10*x + 2 == 0, x]

{{x -> -0.67205}, {x -> -0.277728}, {x -> 0.474889 - 3.23881 I}, {x -> 0.474889 + 3.23881 I}}