**Problem Analysis**

Let's denote the original speed of the train as 'x' km/h and the increased speed as 'x + 5' km/h.
The distance traveled by the train is given as 600 km.
We are given that if the speed is increased by 5 km/h, the journey would take 4 hours less. We can use this information to form an equation.

**Solution**

Let's assume that the original journey takes 't' hours.
The time taken to travel a distance at a constant speed can be calculated using the formula:
Time = Distance/Speed

So, the time taken for the original journey is:
t = 600/x

Now, if the speed is increased by 5 km/h, the time taken would be 't - 4' hours. We can use the same formula to calculate the time:

t - 4 = 600/(x + 5)

Now we have two equations:
t = 600/x
t - 4 = 600/(x + 5)

We can solve these equations to find the value of 'x'.

**Solving the Equations**

Let's solve the equations step by step:

From the first equation, we can write t = 600/x and substitute it into the second equation:

600/x - 4 = 600/(x + 5)

Now, let's cross multiply to eliminate the fractions:

600(x + 5) - 4x(x + 5) = 0

Expanding the equation:

600x + 3000 - 4x^2 - 20x = 0

Rearranging the terms:

4x^2 + 20x - 600x - 3000 = 0

Simplifying:

4x^2 - 580x - 3000 = 0

Now, we can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 4, b = -580, and c = -3000.

Calculating the discriminant:

√(b^2 - 4ac) = √((-580)^2 - 4(4)(-3000)) = √(336400 + 48000) = √384400 = 620

Now, substituting the values into the quadratic formula:

x = (-(-580) ± 620) / (2 * 4)

Simplifying:

x = (580 ± 620) / 8

x = (580 + 620) / 8 or x = (580 - 620) / 8

x = 1200 / 8 or x = -40 / 8

x = 150 or x = -5

Since speed cannot be negative, we discard the negative value.

Therefore, the speed of the train is 150 km/h.