The book by John Bewersdorff is a beginner's book so you can judge my skills. After explaining how all rational roots should be integers if the coefficients are integers, and the Eisenstein criterion, an exercise problem is to reduce the following polynomial to irreducible factors: $$x^6+9x^5+19x^4-4x^3+5x^2-13x-3$$ I am unable to find a 'normal' way to obtain the factors other than assuming generic quadratic-qartic, cubic-cubic, or triple quadratic monic polynomials.
The constant term is the only relaxation in an otherwise cumbersome problem.
I used MATLAB to know the solutions and it so happens that there is a qaurtic polynomial as an irreducible factor. But even that is not visibly irreducible (i.e. Eisenstein's criterion). I might have to verify via $f(x+a)$ if the polynomial is irreducible. Was it supposed to be this tedious an exercise? Any standard route I am unaware of?