The given problem presents a function $f(x) = (x^2 + 2)e^{-x}$ and includes several parts. Let's break down the components of the problem as they are written in Greek:

Γ1. Show that the function $f$ is strictly decreasing and concave.

Γ2. Solve the inequality $f(f'(x) + 5x + 2) > \frac{6}{e^2}$.

Γ3. Show that $f(3x) + f(x) > 2f(2x)$ for all $x > 0$.

Γ4. If $a > 1$, prove that there exists a unique $\beta \in (a, 2a)$ such that $f(3a) + f(a) = 2f(\beta)$.

Would you like help with any specific part of the problem? I can solve each part step-by-step. Let me know if you'd like that!