Completely factor: 45p2 - 120p + 80

I'd suggest we begin by finding the largest factor common to all three terms and then actually completing the factoring:  

5(9p^2 - 24p + 16)            Factors of 16 are 1, 2, 4, 8 and 16.
                                          Factors of 5 are 1 and 5.
Let's try out some "rational roots" made up of one choice from {1, 2, 4, 8, 16} in the numerator and 1 from {1,5} in the denominator. 

Let's solve this using the binomial theorem:

        -(-120) plus or minus sqrt((-120)^2 - 4(45)(80)
p =
                                             90
       120 plus or minus sqrt(14400-14400)
  =
                                        90
         120 plus or minus 0
   =
                     90

    =4/3 and 4/3 (two real, equal roots)

Then the factors are (3x-4) and (3x-4) (two identical factors)

The 3 factors are thus (5), (3x-4) and (3x-4).