BG-MO 2017 Problem 5

Let $n$ be a natural number and $f(x)$ be a polynomial with real coefficients having $n$ different positive real roots. Is it possible the polynomial: $$x(x+1)(x+2)(x+4)f(x)+a$$ to be presented as the $k$ -th power of a polynomial with real coefficients, for some natural $k\geq 2$ and real $a$ ?