KöMaL 2012 A574

Let $n \ge$2 and let $p(x)=x^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0$ be a polynomial with real coefficients. Prove that if for some positive integer $k$ the polynomial ($x$-1)$^{k+1}$ divides $p$($x$) then $\sum_{\ell=0}^{n-1} |a_\ell| > 1+\frac{2k^2}{n}$ .