Use the binomial expansions of x - 1/x :
x - 1/x = 4
(x - 1/x)² = x² - 2 + 1/x² = 16
===> x² + 1/x² = 14
(x - 1/x)³ = x³ - 3x + 3/x - 1/x³
===> x³ - 1/x³ = 3 (x - 1/x) = 12
(x - 1/x)⁴ = x⁴ - 4x² + 6 - 4/x² + 1/x⁴
===> x⁴ + 1/x⁴ = 4 (x² + 1/x²) - 6 = 50
