To solve this equation, we will first expand and simplify the expression:

(X-2)^2 - 8(X-2) + 15 = 0

Expanding (X-2)^2:
X^2 - 4X + 4 - 8(X-2) + 15 = 0

Distribute -8:
X^2 - 4X + 4 - 8X + 16 + 15 = 0

Combine like terms:
X^2 - 12X + 35 = 0

Now we have a quadratic equation in the form of Ax^2 + Bx + C = 0 where A = 1, B = -12, and C = 35. To solve this equation, we can use the quadratic formula:

x = (-B ± √(B^2 - 4AC)) / 2A

Plugging in the values:
x = (12 ± √((-12)^2 - 4135)) / 2*1
x = (12 ± √(144 - 140)) / 2
x = (12 ± √4) / 2
x = (12 ± 2) / 2

x = (12 + 2) / 2 = 14 / 2 = 7
x = (12 - 2) / 2 = 10 / 2 = 5

Therefore, the solutions to the equation are x = 7 and x = 5.