To solve this inequality, we first need to simplify both sides.

Starting with the left side:

| -1 - x^2 |

Since the absolute value of a number will always be non-negative, we can remove the absolute value signs when squared:

= (-1 - x^2)^2

= 1 + 2x^2 + x^4

Now, simplifying the right side:

| 3x - x^2 - 4 |

= |-x^2 + 3x - 4|

= (x^2 - 3x + 4)

Now we have:

1 + 2x^2 + x^4 <= x^2 - 3x + 4

Rearranging and simplifying the inequality:

x^4 + x^2 - x - 3 <= 0

Factoring the left side:

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

The solutions for x are the values that make the inequality true. To find these solutions, we need to consider when the expression is less than or equal to zero:

(x^2 - 2)(x^2 + 1) = 0

x^2 = 2 or x^2 = -1

However, x^2 = -1 has no real solutions, so we consider x^2 = 2:

x = ±√2

Therefore, the solution to the inequality is x ≤ -√2 or x ≥ √2.