4^(x+1)-5*2^(x)+1<0

4^(x + 1) - 5 * 2^(x) + 1 < 0
4^(x + 1) = 4 * 4^x = 4 * 2^2x = 4 * (2^x)^2
4 * (2^x)^2 - 5 * 2^(x) + 1 < 0, t = 2^x
4t^2 - 5t + 1 = 0
D = (-5)^2 - 4 * 4 = 25 - 16 = 9 = 3^2
t1,2 = (5 +- 3)/8 => t1 = 1 = 2^0, t2 = 1/4 = 2^(-2)
4t^2 - 5t + 1 = (t - 1)(t - 1/4), t = 2^x
(2^x - 2^0)(2^x - 2^(-2)) < 0
Решим неравенство методом интервалов: http://bit.ly/2yrRm3k
Ответ: x ∈ (-2; 0)