Question 489050
Regroup by even and odd powers:
X^4+2X^2-8  - (X^3+4X)
2nd term = (-) X(X^2+4), so synthetically divide the 1st term by (X^2+4) in hopes of getting a clean factor (you will).
Then express the whole thing as (X^2+4)*(whatever, collect terms).
The 1st of these has no real roots, but the "whatever" will have.
Hint: 2 of them, simple.
Thus the original equation, known to have 4 roots, will have 2 not real, and the 2 you find from "whatever".