First off, x=0 is a trivial solution.  There cannot be negative solutions.

If we ignore the [.] function we get x^3/24 = x^2 which has solution x=24.  This is also a solution since each of x/2, x/3, x4 will not be rounded down.

The LHS of the equation is bounded above by x^3/24 and below by (x-2)(x-3)(x-4)/24.  These cross the x^2 at x=24 and x=32.216 respectively.

A quick table shows no other solutions in this range.  So the solution set is {0,24}

There is an additional non-integer solution at sqrt(882)=29.7 where the equation becomes 14*9*7=882 (I play slope game )