If (x^2-4x+3)(x^2-3)<0
then one (but not both) of the terms must be less than zero.
Case 1: (x^2-4x+3 < 0) and (x^2-3 > 0)
color(white)("XXXX")x^2-4x+3 < 0
color(white)("XXXX")(x-3)(x-1) < 0
color(white)("XXXX")again one of the terms must be less than zero (and the other greater)
color(white)("XXXX")Since x-3 < x-1
color(white)("XXXX")color(white)("XXXX")x-3 < 0 and x-1 > 0
color(white)("XXXX")color(white)("XXXX")x < 3 and x > 1
color(white)("XXXX")color(white)("XXXX")x epsilon (1,3)
and
color(white)("XXXX")x^2-3>0
color(white)("XXXX")rarr x^2 > 3
color(white)("XXXX")rArr x > sqrt(3)
rArr sqrt(3) < x < 3
Case 2: (x^2-4x+3 > 0) and (x^2-3 < 0)
color(white)("XXXX")x^2-4x+3 > 0
color(white)("XXXX")(x-3)(x-1) > 0
color(white)("XXXX")both terms must be negative or both terms must be positive
color(white)("XXXX")and since x-3 < x-1
color(white)("XXXX")color(white)("XXXX")x-3 > 0 or x-1 < 0
color(white)("XXXX")color(white)("XXXX")rArr x >3 or x < -1
and
color(white)("XXXX")x^2-3 < 0
color(white)("XXXX")rArr x^2 < 3
color(white)("XXXX")rArr -sqrt(3) < x < sqrt(3)
rArr -sqrt(3) < x < -1
Combining: Case 1 or Case 2
-sqrt(3) < x < -1
or
sqrt(3) < x < 3