Equate a circle centered at M (-3; 1) and passing through K (-1: 5).

First, let’s find the radius of this circle.

In the initial data for this task, it is reported that the center of this circle is at point M with coordinates (-3; 1) and passes through point K with coordinates (-1: 5), therefore, the radius of this circle is equal to the length of the KM segment.

Find the length of this segment using the formula for the distance between two points on the coordinate plane:

| KM | = √ ((- 1 – (-3)) ^ 2 + (5 – 1) ^ 2) = √ ((- 1 + 3) ^ 2 + (5 – 1) ^ 2) = √ (2 ^ 2 + 4 ^ 2) = √ (4 + 16) = √20.

Knowing the radius of the circle and its center, we can make the equation for this circle:

(x – (-3)) ^ 2 + (y – 1) ^ 2 = (√20) ^ 2.

Simplifying this equation, we get:

(x + 3) ^ 2 + (y – 1) ^ 2 = 20.

Answer: (x + 3) ^ 2 + (y – 1) ^ 2 = 20.