решение

To solve the inequality \( \log_{x+2}(x^2 - 2) \geq 2 \), follow these steps:

1. Understand the Domain:

   • The base \( x+2 \) must be positive and not equal to 1: \( x+2 > 0 \) and \( x+2 \neq 1 \). Thus, \( x > -2 \) and \( x \neq -1 \).
   • The argument \( x^2 - 2 \) must be positive: \( x^2 - 2 > 0 \). Thus, \( x < -\sqrt{2} \) or \( x > \sqrt{2} \).

2. Rewrite the Inequality: \[ \log_{x+2}(x^2 - 2) \geq 2 \quad \Rightarrow \quad x^2 - 2 \geq (x+2)^2 \]

3. Simplify: \[ x^2 - 2 \geq (x^2 + 4x + 4) \] \[ -2 \geq 4x + 4 \] \[ -6 \geq 4x \] \[ x \leq -\frac{3}{2} \]

4. Find the Intersection of Conditions:

   • From domain: \( x > -2 \) or \( x \leq -\frac{3}{2} \)
   • Intersection is \( -2 < x \leq -\frac{3}{2} \).

5. Solution:

   • The solution to the inequality is \( -2 < x \leq -\frac{3}{2} \).