We can bring all the terms over to the left:
x⁵-x³-8x²+8>0.
Now we can factorise:
x³(x²-1)-8(x²-1)>0, (x³-8)(x²-1)>0,
(x-2)(x²+2x+4)(x-1)(x+1)>0.
The quadratic doesn’t factorise further.
Now we can create our sign table for four intervals:
[2,∞)
|
(1,2)
|
[-1,1]
|
(-∞,-1)
|
+
|
-
|
+
|
-
|
So the inequality is true for [2,∞) and [-1,1].