To solve this problem, we'll be using the formula of a circle in a two-dimensional space. The general form of the equation of a circle is (x-h)² + (y-k)² = r², where (h, k) represents the coordinates of the center of the circle, and 'r' is the radius of the circle.
In this problem, you've been given the coordinates of the center of a circle, which are (-10, -4), and the coordinates of a point on the circle, which are (4, -2). The first step towards finding the equation of the circle is to calculate its radius.
The radius 'r' of a circle can be determined by taking the square root of the sum of the squares of the differences between the X and Y coordinates of a point on the circle and the center of the circle.
Using these coordinates, the radius 'r' is calculated as:
```
r = sqrt((4 - (-10))^2 + ((-2) - (-4))^2)
```
After evaluating the expression, the radius is approximately 14.14213562373095 units.
Now that we have the radius, we can substitute all the values we have (the center and the radius) back into our equation of a circle.
So, the equation of the circle is:
```
(x - (-10))^2 + (y - (-4))^2 = (14.14213562373095)^2
```
After evaluating the expression, and rounding the radius' square to 2 decimal places, the equation of the circle simplifies to:
```
(x - (-10))^2 + (y - (-4))^2 = 200
```
or further simplifies to:
```
(x + 10)^2 + (y + 4)^2 = 200
```
This is the equation of the circle with center at (-10, -4) and a point on the circle at (4, -2).