Does the product of an exponential decrease with a polynomial with positive coefficients have a unique maximum? Consider a function of the form
$$f(x)=\exp(-x)p(x)$$
where $p$ is a polynomial with exclusively positive coefficients. Is it true that $f$ has a unique maximum? It seems like it when playing with functions like this using a graphing calculator. And intuitively the exponential has the longest "spoolup time" but once it starts decreasing faster than the polynomial increases it only ever gets worse.
But I can not prove it. Even nicer would be an explicit representation of this maximum. The first order condition
$$
0 = f'(x) = \exp(-x)[p'(x) - p(x)]
$$
would require $p(x) = p'(x)$. Although this does not look like it should not have a unique solution...

Here are two counter-examples with $2$ local maxima on $\mathbb R^+$, that I found playing around with the coefficients, and trying to keep the polynomial as simple as possible (I focused on $x^n+ax+b$)
$p(x)=10x^3+21x+18$
$q(x)=x^5+40x+20$

The positivity constraint makes it a bit difficult to reverse engineer something like "let solve $p'-p=(x-a_1)(x-a_2)\cdots(x-a_n)$ so that the derivative has many critical points".
Note that the condition on coefficients positivity implies that $p\nearrow$ on $\mathbb R^+$ and since $p(0)\ge 0$ then except maybe in zero, $p$ has not root on $\mathbb R^+$. Therefore the oscillations we see on $p(x)e^{-x}$ results only from the compared growth of the two components and not from $p(x)$ own oscillations.
