eqn1 = 2 b^2 + b^4 + 4 a^3 (3 + b) + 6 a^2 (-1 + 2 b + b^2 - 4 c) +
    4 a (-b^2 + b^3 - 6 b c + 3 c (1 + c)) ==
   4 b^3 + 6 c^2 + 4 b (-3 + c) c^2;(*Óô˵Èʽ½â³öcµÄʵÊý½â£¬ÆäÖÐ0<a<c<a+b<1*)
exp1 = ((-2 + 4 a + 4 b) (-a + c)^3 -
     b (a - c) (3 a^3 + 2 b - 4 b^2 + b^3 + a^2 (6 b - c) - c^3 +
        a (-4 b + 4 b^2 - c^2)))/b;
(*È»ºó½«c´úÈëÉÏÊö±í´ïʽÖеõ½±í´ïʽ¡¾1¡¿*)
exp2 = -(3 + 3 a^4 + 6 a^3 (-2 + b) - 8 b + 6 b^2 - b^3 + 2 a^2 (9 - 11 b + 2 b^2) +
     a (-12 + 24 b - 12 b^2 + b^3));
exp3 = -a (3 a^3 + 6 a^2 b + 4 a (-1 + b) b + b (2 - 4 b + b^2));
(*±í´ïʽ¡¾1¡¿ºÍÉÏÃæÁ½¸ö±í´ïʽÏàµÈ£¬µÃµ½Á½¸ö·½³Ì£¬Çóa,bµÄ½üËÆÖµ(СÊýµãºó±£Áô4λ)ʹµÃ±í´ïʽ¡¾1¡¿Öµ<0*)

cond = {exp1 < 0, 0 < a < c < a + b < 1}

NMinimize[{( exp1 - exp3)^2 + (Subtract @@ eqn1)^2 + (exp1 - exp2)^2, cond}, {a, b, c},
  Method -> "DifferentialEvolution"] // AbsoluteTiming

(* {1.871953, {3.71924*10^-17, {a -> 0.223149, b -> 0.553702, c -> 0.5}}} *)

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