To maintain a constant speed, the bus has to overcome the frictional force acting on it. The frictional force is given by:

$$F_{friction} = \mu F_N$$

Where $$\mu$$ is the coefficient of friction and $$F_N$$ is the normal force acting on the bus. Since the bus is on a flat road, the normal force is equal to the gravitational force:

$$F_N = mg$$

Where $$m$$ is the mass of the bus and $$g$$ is the acceleration due to gravity (approximately $$9.8 \mathrm{~m/s^2}$$).

Substituting the values, we get:

$$F_{friction} = 0.04 \times 500 \times 9.8$$

$$F_{friction} = 196 \mathrm{~N}$$

To maintain a constant speed, the engine must exert a force equal in magnitude to the frictional force. The work done by the engine to overcome the frictional force is given by:

$$W = F_{friction} \times d$$

Where $$d$$ is the distance traveled. First, convert the distance from km to m:

$$d = 4 \mathrm{~km} \times \frac{1000 \mathrm{~m}}{1 \mathrm{~km}} = 4000 \mathrm{~m}$$

Now, calculate the work done:

$$W = 196 \mathrm{~N} \times 4000 \mathrm{~m}$$ $$W = 784000 \mathrm{~J}$$

Convert the work done from joules to kilojoules:

$$W = \frac{784000 \mathrm{~J}}{1000 \mathrm{~J/ kJ}} = 784 \mathrm{~kJ}$$

The work done by the engine of the bus to maintain a speed of 80 km/h for a 4 km distance is $$784.8 \mathrm{~kJ}$$.