Question 392537
Using the rational root test, x=1 and x=2 are zeros of the polynomial, so (x-1)(x-2) or {{{x^2 - 3x + 2}}} divides the polynomial. Dividing {{{(x^4 + 7x^3 - 4x^2 - 52x + 48)/(x^2 - 3x + 2)}}} using long division or synthetic division twice, we get {{{x^2 + 10x + 24}}}, so the polynomial is equal to {{{(x^2 - 3x + 2)(x^2 + 10x + 24)}}}. The first expression has roots at 1 and 2 (we've already figured this), and the other expression factors to (x+6)(x+4) so it has roots at -6 and -4. Therefore the four roots are 1,2, -4, -6 and the factorization is


{{{(x-1)(x-2)(x+4)(x+6)}}}