Question 139
Let $n$ be a positive integer and $f(x)$ be a polynomial of degree $n$ with $n$ distinct real positive roots. Are there a positive integer $k \ge 2$ and a real polynomial $g(x)$ such that
$$
x(x+1)(x+2)(x+4)f(x) + 1 = (g(x))^k?
$$
$$
x(x+1)(x+2)(x+4)f(x) + 1 = (g(x))^k?
$$