First, factor the numerator and denominator of the expression as:
#((x -3)(x + 4))/((2x + 3 )(x - 3))#
Next, cancel the common terms from the numerator and denominator:
#(color(red)(cancel(color(black)((x -3))))(x + 4))/((2x + 3 )color(red)(cancel(color(black)((x -3))))) =>#
#(x + 4)/(2x + 3)#
However, because we cannot divide by #0# we must ensure:
#2x + 3 != 0# and #x - 3 != 0#
Or
Condition 1:
#2x + 3 != 0#
#2x + 3 - color(red)(3) != 0 - color(red)(3)#
#2x + 0 != -3#
#2x != -3#
#(2x)/color(red)(2) != -3/color(red)(2)#
#(color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) != -3/2#
#x != -3/2#
Condition 2:
#x - 3 != 0#
#x - 3 + color(red)(3) != 0 + color(red)(3)#
#x - 0 != 3#
#x != 3#
Therefore, the simplified expression is:
#(x + 4)/(2x + 3)# Where #x != -3/2# and #x != 3#
