What is the solution for x^3 + 5x^2 -x - 5

We need to find the solution for x^3 + 5x^2 -x - 5
< 0.

x^3 + 5x^2 -x - 5 <
0

=> x^2( x + 5) - 1(x + 5) <
0

=> (x^2 - 1)(x + 5) <
0

=> (x - 1)(x + 1)(x + 5) <
0

For the expression on the left to be less than 0, either
all three factors have to be less than 0 or only one of them is less than
0.

If all three are less than
0,

=> x - 1 < 0 , x + 1 < 0 and x + 5
< 0

=> x < 1 , x < -1 and x
< -5

x < -5 satisfies all the three
conditions. Or x lies in (-inf , -5).

If only one of them
is less than 0,

x - 1 < 0 , x + 1 > 0 and x +
5 > 0

=> x < 1 , x > -1 and x
> -5

This gives the values of x in the set ( -1 ,
1)

x - 1 > 0 , x + 1 < 0 and x + 5 >
0

=> x > 1 , x < -1 and x > -5
, no valid solution

x - 1 > 0 , x + 1 > 0 and
x + 5 < 0

=> x > 1 , x > -1 and
x < -5, no valid solution

The required
values lie in x = (-inf.,-5) U (-1, 1).