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).
