To find the roots of the given equation, we can set each factor equal to zero and solve for x:

From (x-1)^2(x-3) = 0:
(x-1)^2 = 0
x-1 = 0
x = 1

x-3 = 0
x = 3

Therefore, the roots for this part of the equation are x = 1 and x = 3.

From (x-4)^2(x-3) = 0:
(x-4)^2 = 0
x-4 = 0
x = 4

x-3 = 0
x = 3

Therefore, the roots for this part of the equation are x = 4 and x = 3.

Therefore, the roots of the original equation (x-1)^2(x-3)(x-4)^2(x-3) = 0 are x = 1, x = 3, and x = 4.