KöMaL 2009 A480

Let $p$($z$) be a complex polynomial of degree $n$, and suppose that all (complex) roots of $p$($z$) are of unit modulus. For every real number $c \ge$0, show that the roots of the polynomial 2$z$($z$-1)$p$'($z$)+ ( ($c$-$n$)$z$+($c$+$n$) ) $p$($z$) are of unit modulus as well.