Hi Cedric,

it would be useful if you could attach your notebook an/or use the equation editor when you post. In your inline code there appear to be a couple of syntactical errors. For example if you want to evaluate an equation you should use double equal signs. Also "ax" is not the same as "a x"; "ax" is one variable called "ax" and "a x" is the product of the variables a and x. Now, guessing a bit what you might have wanted to solve I used:

Solve[16 x^4 - 40 a x^3 + (15 a^2 + 24 b) x^2 - 18 a b x + 3 b^2 == 0 && 5 a x - 4 x^2 - b > 0 && 15 a x - 20 x^2 - 3 b < 0 && 
4 x^3 - 8 c x^2 + 5 a c x - c b < 0 && a < 0 && x < 0 && c < 0, x]

which gives:

So there are sill some Root-objects in there. If you use Reduce instead of Solve you get another variation of the solution. Individual solutions can be found using FindInstance, but that won't solve your problem:

FindInstance[
 16 x^4 - 40 a x^3 + (15 a^2 + 24 b) x^2 - 18 a b x + 3 b^2 == 0 && 
  5 a x - 4 x^2 - b > 0 && 15 a x - 20 x^2 - 3 b < 0 && 
  4 x^3 - 8 c x^2 + 5 a c x - c b < 0 && a < 0 && x < 0 && c < 0, {x, a, b, c}]

which gives:

{{x -> -1, a -> -2, b -> 1/3 (6 + 4 Sqrt[3]), c -> -1}}

Hope this helps.

Cheers,

Marco