Resolve a conditional expression including Exists and ForAll

Question

I don't know the Resolve function exactly, especially when involved with Exists and ForAll, could you please help me in the following example?

$\forall C$, can we find some $x>0$, such that $$ c(x)=(C+2x)x^2+C^2x+2Cx^2+C\geq0, $$ and $$ C<-2x $$ holds at the sametime?

Answer 1

Exists[x,  x > 0 && ForAll[c, c < -2 x && (c + 2 x) x^2 + c^2 x + 2 c x^2 >= 0]]
Resolve[%]

(* False *)

Edit: In thinking about this, I perhaps took the question as stated too literally and that's not what you meant, i.e., it is obvious that it is not true that for all C, C <-2 X, which the above shows. However, if you meant (and I think this true) for all C with the condition C < -2 X, that's done this way:

Exists[x, x > 0, ForAll[c, c < -2 x, (c + 2 x) x^2 + c^2 x + 2 c x^2 >= 0]]
Reduce[%]

(* True *)

You can use FindInstance to get instance(s), like:

FindInstance[x > 0 && c < -2 x && (c + 2 x) x^2 + c^2 x + 2 c x^2 >= 0, {x, c}, Reals]

(*  {{x -> 1, c -> -4}}  *)