If #(x-a)# is a factor of #f(x)=4x^3-9x^2+6x+1#, #a# could be a factor of #+-1/4# - here #1# comes from constant term and #4# comes from the coefficient of highest power #x^3#.
Hence, #a# could be #+-1/4#, #+-1/2# or #+-1#.
Further from factor theorem if #(x-a)# is a factor of #f(x)# then #f(a)=0#.
We know here that #f(1)=2# and #f(-1)=-18# and hence #(x-1)# and #(x+1)# are not the factors of #f(x)#. Similarly
#f(1/2)=4/8-9/4+3+1!=0# and #f(-1/2)=-4/8+9/4-3+1!=0#
#f(1/4)=4/64-9/16+3/4+1!=0# and#f(-1/4)=-4/64-9/16-3/4+1!=0#
Hence #f(x)=4x^3-9x^2+6x+1# does not have rational factors.