4^(Х+1)-6^х-2*9^(Х+1)=0

4 * (2^x)^2 - 2^x * 3^x - 2 * 9 * (3^x)^2 = 0

Разделим на (3^x)^2 > 0:
4 * (2^x / 3^x)^2 - 2^x / 3^x - 18 = 0

Сделаем замену t = (2/3)^x > 0.
4t^2 - t - 18 = 0
D = 1 + 4 * 4 * 18 = 1 + 288 = 289 = 17^2
t = (1 ± 17)/(2 * 4)
t1 = (1 + 17)/8 = 9/4
t2 = (1 - 17)/8 < 0 - не подходит

Возвращаемся к x:
(2/3)^x = 9/4
(2/3)^x = (2/3)^(-2)
x = -2

Ответ. x = -2.