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#
