To solve this equation, we need to first expand the left side of the equation:

(x-3)(x-4)(x-5)(x-6) = 1680
(x^2 - 4x - 3x + 12)(x^2 - 5x - 6x + 30) = 1680
(x^2 - 7x + 12)(x^2 - 11x + 30) = 1680
x^4 - 11x^3 + 30x^2 - 7x^3 + 77x^2 - 210x + 12x^2 - 132x + 360 = 1680
x^4 - 18x^3 + 119x^2 - 342x + 360 = 1680

Next, we will set the equation equal to zero:
x^4 - 18x^3 + 119x^2 - 342x + 360 - 1680 = 0
x^4 - 18x^3 + 119x^2 - 342x - 1320 = 0

Now, we can factor the equation or use numerical methods to find the solutions.