# Find all values of the paramaters $m$ for which the equation has four roots smaller 2

I want to find all values of the paramaters $m$ for which the equation $$t^4 + (2 m - 1) t^2 - 18 m - 72=0$$ has four real distinct solutions less than 2. Put $t = x^2$, I tried

Clear[f];
f[x_] := x^2 + (2 m - 1) x - 18 m - 72;
d := Discriminant[f[x], x];
sol = Solve[f[x] == 0, x];
x1 = x /. sol[[1]];
x2 = x /. sol[[2]];
Reduce[{d > 0, x1 > 0, x2 > 0, x2 < 4}, m]


-6 < m < -4

How can I reduce my code?

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You can notice that t==-3, t==3 are always solutions, so not all four solutions will be less than 2.

Factor[t^4 + (2 m - 1) t^2 - 18 m - 72]
(* (-3 + t) (3 + t) (8 + 2 m + t^2) *)


So the problem reduces to the second degree polynomial :

Reduce[Less @@ Join[Solve[8 + 2 m + t^2 == 0, t][[All, 1, 2]], {2}], m]
(* -6 < m < -4 *)


Update :

With[{sols = Solve[t^4 + 2 (m + 1) t^2 + m + 7 == 0, t][[All, 1, 2]]},
Reduce[Join[Unequal @@@ Subsets[sols, {2}], # < 2 & /@ sols], m, Reals]
]
(* -(31/9) < m < -3 *)

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How about with equation t^4 + 2 (m + 1) t^2 + m + 7==0? – minthao_2011 Jan 15 '14 at 7:53