To solve this circuit using mesh analysis, we'll go through the following steps:

Step 1: Label Mesh Currents

In this circuit, we have four loops where we can define the mesh currents:

1. $i_1$ in the left loop
2. $i_2$ in the upper middle loop
3. $i_3$ in the lower middle loop
4. $i_4$ in the right loop

Step 2: Apply Kirchhoff's Voltage Law (KVL) to Each Loop

We will write KVL equations for each loop based on the components and voltages in each loop. Each mesh equation will sum to zero, taking the passive sign convention into account (voltage drops are positive in the direction of current flow).

Mesh Equations

1. Loop 1 (Left loop with $i_1$): $-3A \cdot 5\Omega - 10\Omega(i_1 - i_2) - 15\Omega(i_1 - i_3) + 4V = 0$

2. Loop 2 (Upper middle loop with $i_2$): $-20\Omega(i_2 - i_1) - 5\Omega(i_2 - i_4) - 25\Omega(i_2 - i_3) = 0$

3. Loop 3 (Lower middle loop with $i_3$): $-6V - 15\Omega(i_3 - i_1) - 25\Omega(i_3 - i_2) - 5\Omega(i_3 - i_4) = 0$

4. Loop 4 (Right loop with $i_4$): $-2A \cdot 40\Omega - 5\Omega(i_4 - i_2) - 5\Omega(i_4 - i_3) - 10\Omega = 0$

Step 3: Solve the Equations

These equations can now be solved simultaneously to find the values of $i_1$, $i_2$, $i_3$, and $i_4$. For brevity, I will provide the setup of the simultaneous equations in matrix form: