Introduce new variable  y = x-3  (the central point for the sequence x, x+2, x+4, x+6).


Then your equation takes the form

(y-3)*(y-1)*(y+1)*(y+3) = -16,   or

 = -16,

 = 0,

 = 0,


which implies   = 5,  and hence  y = +/- .


Answer.  The original equation has the solutions  x =  and  , each of multiplicity 2.

    The problem has a hidden symmetry around the central point.
    When written at variables centered at this point, the symmetry becomes explicit and facilitates the problem' solution.