X^3-8/x^2+2x+4>=3

Interpret the inequality as (x^3 - 8) / (x^2 + 2x + 4) >= 3.

• Factor the numerator: x^3 - 8 = (x - 2)(x^2 + 2x + 4).
• Note the denominator: x^2 + 2x + 4 = (x + 1)^2 + 3 > 0 for all real x, so the fraction simplifies exactly to x - 2 for all real x.

Thus the inequality becomes: x - 2 >= 3 => x >= 5.

Solution: x ∈ [5, ∞).