The provided image contains a multi-part mathematical problem that involves analysis of functions and their properties. I’ll break it down step by step.

Part (I):

Three functions are defined:

• $f(x) = -x^2 + 2x - 2$,
• $g(x) = \frac{-2x - 6}{2x + 1}$,
• $h(x) = \sqrt{x - 2}$.

Tasks:

1. Determine $D_f$, $D_g$, $D_h$: The domains of the functions.
2. Variation tables:
   • Draw the variation tables for $f, g, h$.
3. Intersection points and graphing:
   • Nature of $(C_f)$ and $(C_g)$, their intersections, and graphing considerations.
   • Solve $f(x) \geq g(x)$ graphically.
   • Solve the equation $\frac{-2x - 6}{2x + 1} = m$, discussing parameter $m$.

Part (II):

Composite functions:

1. Determine $D_{f \circ h}, D_{h \circ g}, f \circ h \circ g$.
2. Analyze monotonicity of $f \circ h$ over intervals $[2, 3]$ and $[3, +\infty[$.

Part (III):

Function $F$:

Defined piecewise: