Let #f(x) = x^4-2x^3-12x^2+18x+27#
By the rational roots theorem, any rational roots of #f(x) = 0# must be of the form #p/q# where #p, q# are integers, #q != 0#, #p# a divisor of #27# and #q# a divisor of #1#.
So the only possible rational roots are:
#+-1#, #+-3#, #+-9#, #+-27#
#f(1) = 1-2-12+18+27 = 32#
#f(-1) = 1+2-12-18+27 = 0#
#f(3) = 81-54-108+54+27 = 0#
#f(-3) = 81+54-108-54+27 = 0#
So #x=-1#, #x=3# and #x=-3# are roots of #f(x) = 0# and #(x+1)#, #(x-3)# and #(x+3)# are factors of #f(x)#.
The remaining factor must be #(x-3)# in order that when multiplied by the other factors the coefficient of the #x^4# term is #1# and the constant term #27#.
graph{x^4-2x^3-12x^2+18x+27 [-10, 10, -5, 5]}
