Question

Solve completely \large x^4-4=x^3+2x

Answer 1

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).