On occasion, a factorization is not easily manually obtained.  Note that the primes of each term {1,9,23,15} are {1,3,5} and since 23 is itself a prime, it can only be generated by added terms.  23x is divided in "8x+15x" and "-9x^2" is divided into "-8x^2-x^2". 

x^3 - 8x^2 + 15x - x^2 + 8x - 15

when (x-1) is factored out, the intermediate factors are:

(x^2 - 8x + 15)(x-1)

When "x^2 - 8x + 15" is isolated, the same logic is applied, dividing "-8x" into "-5x-3x":