Move everything to one side and factorize by grouping.
x⁴ - 4 = x³ + 2x
x⁴ - x³ - 2x - 4 = 0
(x⁴ - 4) - (x³ + 2x) = 0
(x² - 2) (x² + 2) - x (x² + 2) = 0
(x² - x - 2) (x² + 2) = 0
(x - 2) (x + 1) (x² + 2) = 0
Then
x - 2 = 0 or x + 1 = 0 or x² + 2 = 0
x = 2 or x = -1 or x² = -2
If x is real, then the third case has no solution. Then x = 2 or x = -1. Otherwise, we can solve over the complexes and include x = ± √2 i where i = √(-1).
