x = 1/8 (a1^4 E^(4 I [Tau] [Omega]) - 2 a1^2 E^(  2 I [Tau] [Omega]) (b2 Sqrt[1 - t] + b1 Sqrt[t])^2 + (b2 Sqrt[ 1 - t] + b1 Sqrt[t])^4);

Expand[x]

b2^4/8 - 1/4 a1^2 b2^2 E^(2 I [Tau] [Omega]) +  1/8 a1^4 E^(4 I [Tau] [Omega]) + 1/2 b1 b2^3 Sqrt[1 - t] Sqrt[t] - 1/2 a1^2 b1 b2 E^(2 I [Tau] [Omega]) Sqrt[1 - t] Sqrt[t] +  3/4 b1^2 b2^2 t - (b2^4 t)/4 - 1/4 a1^2 b1^2 E^(2 I [Tau] [Omega]) t +  1/4 a1^2 b2^2 E^(2 I [Tau] [Omega]) t +  1/2 b1^3 b2 Sqrt[1 - t] t^(3/2) - 1/2 b1 b2^3 Sqrt[1 - t] t^(3/2) + ( b1^4 t^2)/8 - 3/4 b1^2 b2^2 t^2 + (b2^4 t^2)/8

1/2 b1 b2^3 Sqrt[1 - t] Sqrt[t] - 1/2 b1 b2^3 Sqrt[1 - t] t^(3/2) =  1/2 b1 b2^3 (1 - t)^(3/2) Sqrt[t]

1/8 a1^4 E^(4 I [Tau] [Omega]) -  1/4 a1^2 b1^2 E^(2 I [Tau] [Omega]) t -  1/2 a1^2 b1 b2 E^(2 I [Tau] [Omega]) Sqrt[1 - t] Sqrt[t] - 1/4 a1^2 b2^2 E^(2 I [Tau] [Omega]) +  1/4 a1^2 b2^2 E^(2 I [Tau] [Omega]) t + (b1^4 t^2)/8 +  1/2 b1^3 b2 Sqrt[1 - t] t^(3/2) + 3/4 b1^2 b2^2 t -  3/4 b1^2 b2^2 t^2 + 1/2 b1 b2^3 Sqrt[1 - t] Sqrt[t] -  1/2 b1 b2^3 Sqrt[1 - t] t^(3/2) + b2^4/8 - (b2^4 t)/4 + (b2^4 t^2)/8

1/8 a1^4 E^(4 I [Tau] [Omega]) - 1/4 a1^2 b1^2 E^(2 I [Tau] [Omega]) t - 1/2 a1^2 b1 b2 E^(2 I [Tau] [Omega]) Sqrt[1 - t] Sqrt[t] +1/4 a1^2 2^2 E^(2 I [Tau] [Omega]) (-1 + t) + (b1^4 t^2)/8 +  1/2 b1^3 b2 Sqrt[1 - t] t^(3/2) + -(3/4) b1^2 b2^2 (-1 + t) t +  1/2 b1 b2^3 (1 - t)^(3/2) Sqrt[t] + 1/8 b2^4 (-1 + t)^2