Explanation:
4/x + 4 = x - 2/x
multiply everything by x
4 + 4x = x² - 2
x² - 4x - 6 = 0
the general solution to such a quadratic equation
ax² + bx + c = 0
is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
x = (4 ± sqrt((-4)² - 4×1×-6))/(2×1) =
= (4 ± sqrt(16 + 24))/2 = (4 ± sqrt(40))/2 =
= (4 ± sqrt(4×10))/2 = (4 ± 2×sqrt(10))/2 =
= 2 ± sqrt(10)
x1 = 2 + sqrt(10) = 5.16227766...
x2 = 2 - sqrt(10) = -1.16227766...