How to Factor a Fifth Degree Polynomial?

Let p(x) = x^5 - 5x^4 - x^3 + x^2 + 4 . The rational roots theorem says the only possible rational roots are divisors of 4 or their negatives, i.e. 1, -1, 2, -2, 4, -4. Check that p(1) = 0 and p(-1) = 0. That means (x-1) and (x+1) are factors. Hence (x^2-1) is a factor and the other factor is p(x) / (x^2-1) which you can obtain by synthetic division, or by long division. That "other factor" is a cubic. In this example the rational roots theorem shows it has no rational roots and (because it is a cubic) must therefore be irreducible. Call that polynomial q(x). Then p(x) = (x-1)(x+1)q(x) is the desired factorization. I leave for you to figure out what q(x) is.