Introduction:
In this problem, we are given that there are four numbers in arithmetic progression (AP) whose sum is 24 and their product is 945. We need to find these four numbers.

Step 1: Finding the common difference:
Let's assume the first term of the AP as 'a' and the common difference as 'd'. Since the numbers are in AP, the second term would be 'a + d', the third term would be 'a + 2d', and the fourth term would be 'a + 3d'.

Step 2: Formulating equations:
Using the above assumptions, we can form the following equations based on the given information:
1. Sum of the four numbers: a + (a + d) + (a + 2d) + (a + 3d) = 24
2. Product of the four numbers: a * (a + d) * (a + 2d) * (a + 3d) = 945

Step 3: Solving the equations:
Solving the first equation, we get:
4a + 6d = 24
2a + 3d = 12 (dividing by 2)

Solving the second equation, we get:
a * (a + d) * (a + 2d) * (a + 3d) = 945
a * (a + 3d) * (a + 2d) * (a + d) = 945
(a^2 + 3ad)(a^2 + 3ad + 2ad + 6d^2) = 945
(a^2 + 3ad)(a^2 + 5ad + 6d^2) = 945
(a^2 + 3ad)(a^2 + 5ad + 6d^2) - 945 = 0

Step 4: Solving the quadratic equation:
We can solve the quadratic equation obtained in the previous step using the quadratic formula or factoring. Let's assume the roots of the quadratic equation are 'm' and 'n'.

a^2 + 3ad = m (Equation 1)
a^2 + 5ad + 6d^2 = n (Equation 2)

Solving these equations simultaneously, we get:
a = (m - 3d) (from Equation 1)
a = (n - 5d - 6d^2) (from Equation 2)

Setting the values of 'a' obtained from both equations equal, we get:
m - 3d = n - 5d - 6d^2

Simplifying the equation, we get:
6d^2 - 2d - (m - n) = 0

Using the quadratic formula, we can find the values of 'd'.

Step 5: Finding the values of 'a' and 'd':
Once we have the values of 'd', we can substitute them back into the equations to find the values of 'a'.

Step 6: Finding the four numbers:
Using the values of 'a' and 'd' obtained in the previous step,

3, 5, 7, 9