Write the equation of a circle centered at point A (-3; 2) passing through point B (0; -2).

According to the condition of the problem, this circle must pass through the current B with coordinates (0; -2), and the center of this circle is at point A with coordinates (-3; 2).

Therefore, the radius of this circle is equal to the distance between points A and B.

To find this distance, we use the formula for the distance between two points A and B on the coordinate plane with coordinates A (x1; y1) and B (x2; y2):

| AB | = √ ((x1 – x2) ² + (y1 – y2) ²).

In this case, x1 = -3, y1 = 2, x2 = 0, y2 = -2.

Substituting these values ​​into the formula for the distance between points A and B, we get:

| AB | = √ ((- 3 – 0) ² + (2 – (-2)) ²) = √ ((- 3) ² + (2 + 2) ²) = √ (3² + 4²) = √ (9 + 16 ) = √25 = 5.

Therefore, the radius of this circle is 5.

Write down the equation of the circle
It is known that the equation of a circle of radius R centered at the point O (x0; y0) has the following form:

(x – x0) ² + (y – y0) ² = R².

Therefore, we can write the equation of a circle of radius 5 centered at point B (0; -2):

(x – 0) ² + (y – (-2)) ² = 5²,

or after simplification:

x² + (y + 2) ² = 25.

Checking the results
Make sure that the circle given by the equation x² + (y + 2) ² = 25 passes through point A (-3; 2).

Substituting the values ​​x = -3 and y = 2 into the equation of the circle, we get:

(-3) ² + (2 + 2) ² = 25;

3² + 4² = 25;

9 + 16 = 25;

25 = 25.

We got the correct identity, therefore, the circle given by the equation x² + (y + 2) ² = 25 passes through the point A (-3; 2).

Answer: the required equation of the circle is x² + (y + 2) ² = 25.