Prove that for any integer $d>3$, polynomial $\displaystyle p(x)=x^{d}+(-2d+1)x^{d-1}+\sum_{k=0}^{d-2}(k^2+k+1)x^{k}$ is irreducible.
This is another problem from the sheet given to me by my teacher. I was hoping for an application of Perron's Criterion but nothing came out of it.