Does this polynomial admit more than a single real root?

Consider the polynomial $p(x)$ such that:

$$ p(x) = 1 + 2x + 3x^2 + 4x^3 + \cdots + (2n+2)x^{2n+1} $$where $ n $ is a natural number. Because $p(x)$ has odd degree, then from the Complex conjugate root theorem it follows that it has at least one real root. My question is then, is this root unique ? This question is actually related to this other question. I plotted the graph for $ n = 0 $ to $ n = 7 $, and in all cases the polynomial has a single real root.