Math Olympiad | Nice Exponential Problem | Find the Value x = ?

[@RyanLewis-Johnson-wq6xs]

(16^x+(2^x+1)^4+1)/(4^x+(2^x+1)^2+1)=2 x=-Log[2,1+Sqrt[5]]=Log[((Sqrt[5]-1)/2),2]

[@satrajitghosh8162]

Writing 2^ x = a one gets a^4 + ( a + 1)^4 + 1 = 2 ( a^2 + (a + 1)^2 + 1) Again a^4 + ( a + 1)^4 + 1 = 2 ( a ^4 + 2 a^3 + 3 a ^2 + 2 a + 1) = 2 ( a^2 + a + 1)^2 This implies 2 ( a^2 + a + 1)^2 = 2 ( a^2 + (a + 1)^2 + 1) = 4 (a^2 + a + 1) Hereby a^2 + a + 1 = 2 ( a + 2)( a - 1)= 0. 2 ^ x = a = -2, 1