Let's denote the original speed as "x" miles per hour.

When the speed is increased by 4 miles per hour, the new speed becomes "x + 4" miles per hour.

We are given that when the speed is increased by 4 miles per hour, it takes 4 hours less to cover a distance of 32 km. We can set up the equation using the formula:

Time = Distance / Speed

Original time = 32 km / x New time = 32 km / (x + 4)

According to the given information, the new time is 4 hours less than the original time. So, we can write:

Original time - New time = 4 hours

32 km / x - 32 km / (x + 4) = 4

To solve this equation, let's first find a common denominator, which is x(x + 4):

(32(x + 4) - 32x) / (x(x + 4)) = 4

Now, simplify the equation:

(32x + 128 - 32x) / (x(x + 4)) = 4

128 / (x(x + 4)) = 4

Now, cross-multiply:

128 = 4(x(x + 4))

Divide both sides by 4:

32 = x(x + 4)

Now, let's solve for x by factoring:

x(x + 4) = 32

x² + 4x - 32 = 0

Now, we can factor the quadratic equation:

(x + 8)(x - 4) = 0

Setting each factor equal to zero:

x + 8 = 0 or x - 4 = 0

If x + 8 = 0, then x = -8, but since we're looking for a speed, it must be positive.

If x - 4 = 0, then x = 4.

So, the previous speed (original speed) was 4 km per hour.