To solve the inequality, we need to consider two cases:

Case 1: x + 5 > 0
If x + 5 > 0, then x > -5. In this case, the absolute value function can be written as x + 5.
So, the inequality becomes x + 5 + 6 > 0
x + 11 > 0
x > -11

Therefore, if x > -5, the inequality x + 5 + 6 > 0 is satisfied for all x > -11.

Case 2: x + 5 < 0
If x + 5 < 0, then x < -5. In this case, the absolute value function becomes -(x + 5) = -x - 5.
So, the inequality becomes -x - 5 + 6 > 0
-x + 1 > 0
-x > -1
x < 1

Therefore, if x < -5, the inequality -x - 5 + 6 > 0 is satisfied for all x < 1.

Combining both cases, the solution to the inequality X * | x+ 5 | + 6 > 0 is x < 1 or x > -5.