According to the ration root theorem the roots of the polynomial would be out of the factors of #5/3# which could be #+- 1.+-5,+-1/3#. By trial and error,it can be ascertained that x=-1 and x=5 are the roots of the polynomial. That implies that (x+1)(x-5) would be factors of the polynomial. Now divide the polynomial by the product of (x+1)(x-5) that is # x^2 -4x-5#.
The division would give a quotient #3x^2 +2x-1# leaving 0 remainder. Using the remainder theorem again, it can be ascertained that x=-1 is a root of #3x^2 +2x-1#. Therefore x+1 would be one of its factors. The other factor can be had by dividing #3x^2+2x-1# with x+1. The other factor would be 3x-1.
All the factors of the polynomial would thus be #(x+1)^2 (x-5)(3x-1)#
