**Stable and Unstable Equilibrium in the context of Point Charges**

In order to determine whether the charges 4q, -q, and 2q are in stable or unstable equilibrium, we need to analyze the forces acting on each charge and examine their respective positions on the x-axis.

**Understanding Equilibrium:**

Equilibrium refers to a state where the net force acting on an object is zero. When an object is in equilibrium, it can either be in a stable or unstable equilibrium, depending on its response to infinitesimal displacements.

- **Stable Equilibrium:** If a system returns to its original position after being displaced slightly, it is said to be in stable equilibrium.
- **Unstable Equilibrium:** If a system moves further away from its original position when displaced slightly, it is said to be in unstable equilibrium.

**Analyzing the Charges and their Positions:**

We have three charges: 4q, -q, and 2q, placed at points x=0 and x=2a on the x-axis, respectively.

- Charge 4q is located at x=0.
- Charge -q is not specified with a location, so we assume it is also at x=0.
- Charge 2q is located at x=2a.

**Calculating Forces:**

To determine the forces acting on each charge, we consider Coulomb's Law, which states that the force between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.

- The force between 4q and -q is given by F1 = k * (4q) * (-q) / (r^2), where r is the distance between them.
- The force between 4q and 2q is given by F2 = k * (4q) * (2q) / ((2a)^2).
- The force between -q and 2q is given by F3 = k * (-q) * (2q) / ((2a)^2).

**Analysis of Forces and Equilibrium:**

- If the forces F1, F2, and F3 are in such a way that they balance each other, then the system is in equilibrium.
- If the forces do not balance each other, the system is not in equilibrium.

To determine the equilibrium condition, we need to calculate the net force acting on each charge:

- For charge -q, the net force is F1 + F3.
- For charge 2q, the net force is F2 - F3 (since F3 is acting in the opposite direction).

**Conclusion:**

By calculating the net forces on each charge and analyzing their positions, we can determine whether the charges are in stable or unstable equilibrium.

It is important to note that the distances between the charges and the magnitudes of the charges are crucial factors in determining the equilibrium condition. Without specific values for these variables, it is not possible to provide a definitive answer regarding the stability of the equilibrium.