To solve this equation, we need to expand both sides and then simplify.

Expanding the left side:
(x-4)(x-5)(x-6) = (x^2 - 5x - 4x + 20)(x-6)
= (x^2 - 9x + 20)(x-6)
= x^3 - 6x^2 - 9x^2 + 54x + 20x - 120
= x^3 - 15x^2 + 74x - 120

Expanding the right side:
(x-2)(x-5)(x-6) = (x^2 - 5x - 2x + 10)(x-6)
= (x^2 - 7x + 10)(x-6)
= x^3 - 6x^2 - 7x^2 + 42x + 10x - 60
= x^3 - 13x^2 + 52x - 60

So, the equation becomes:
x^3 - 15x^2 + 74x - 120 = x^3 - 13x^2 + 52x - 60

Now, we can simplify by subtracting x^3 and 52x from both sides:
-15x^2 + 74x - 120 = -13x^2 + 52x - 60

Now, we can subtract -13x^2 and 52x from both sides:
-15x^2 + 74x - 120 = -13x^2 + 52x - 60
-15x^2 + 74x - 120 + 13x^2 - 52x + 60 = 0
-2x^2 + 22x - 60 = 0

Now, we have a quadratic equation that we can solve using the quadratic formula. Let's use the quadratic formula to find the solutions for x:

x = (-b ± √(b^2 - 4ac))/(2a)
where a = -2, b = 22, and c = -60

Plugging in the values:
x = (-22 ± √(22^2 - 4(-2)(-60)))/(2(-2))
x = (-22 ± √(484 - 480))/(-4)
x = (-22 ± √4)/(-4)
x = (-22 ± 2)/(-4)

Now, we have two possible solutions for x:

x = (-22 + 2)/(-4) = -20/-4 = 5x = (-22 - 2)/(-4) = -24/-4 = 6

Therefore, the solutions to the equation are x = 5 and x = 6.