Given information:
The equation of the given circle is X^2 - Y^2 - 6X + 12Y + 15 = 0.

Explanation:
To find the equation of a circle that is concentric with the given circle and has double its radius, we need to follow these steps:

Step 1: Determine the center of the given circle.
To determine the center of the given circle, we need to convert the given equation into the standard form of a circle equation, which is (X - h)^2 + (Y - k)^2 = r^2. In this form, (h, k) represents the coordinates of the center of the circle.

Step 2: Find the radius of the given circle.
The radius of a circle can be determined by taking the square root of the coefficient of the squared terms. In this case, the coefficient of X^2 is 1, and the coefficient of Y^2 is -1. Therefore, the radius of the given circle is √1 = 1.

Step 3: Determine the radius of the new circle.
Since the new circle has double the radius of the given circle, the radius of the new circle will be 2 times the radius of the given circle. Therefore, the radius of the new circle is 2 * 1 = 2.

Step 4: Determine the equation of the new circle.
Since the new circle is concentric with the given circle, it will have the same center. The radius of the new circle is 2. Therefore, the equation of the new circle can be written as (X - h)^2 + (Y - k)^2 = 2^2.

Step 5: Substitute the coordinates of the center of the given circle into the equation of the new circle.
We need to find the coordinates (h, k) of the center of the given circle and substitute them into the equation of the new circle. The coordinates (h, k) can be determined by completing the square for X and Y terms separately.

Step 6: Complete the square for X terms.
To complete the square for the X terms, we need to add and subtract the square of half the coefficient of X from the given equation. The coefficient of X is -6, so we add and subtract (-6/2)^2 = (-3)^2 = 9.

The equation becomes:
(X^2 - 6X + 9) - 9 - Y^2 + 12Y + 15 = 0

Simplifying further:
(X - 3)^2 - 9 - Y^2 + 12Y + 15 = 0
(X - 3)^2 - Y^2 + 12Y + 6 = 0

Step 7: Complete the square for Y terms.
To complete the square for the Y terms, we need to add and subtract the square of half the coefficient of Y from the previous equation. The coefficient of Y is 12, so we add and subtract (12/2)^2 = (6)^2 = 36.

The equation becomes:
(X - 3)^2 - Y^2 + 12Y + 6 - 36 + 36 = 0