solve each inequality:
(x-4)(x-2)^2(x-3)^2 < 0
Had this been an equation the solutions would have been
x = 4,2, and 3
so on the number line you have the following segments to consider
a) less than 2
b) between 2 and 3
c) between 3 and 4
d) greater than 4
Of course the (x-2)^2(x-3)^2 is always positive, because squaring something results in a positive or at worst zero, but you want <0, so to have the entire result to be negative, x-4 has to be negative
that is true for all x < 4
so your solution is x < 4