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.