Roots of polynomial/exponential function Let $n\in \mathbb{N}$ and $\sigma \in (0,1]$ be fixed.
I am interested in finding the roots of the function
\begin{equation}
f(x) = \big(x^2-n^2-\sigma n^2 x\big)e^{x} - \big(x^2-n^2+\sigma n^2 x\big)e^{-3x};
\end{equation}
on $[0,n^2]$, namely solving $f(x) = 0$ for $x \in [0,n^2]$.
I would be happy with an approximation formula, which clearly indicates the dependence on $n \in \mathbb{N}$.
Clearly one root is $x=0$.
A way to proceed for finding the other root(s?) could of course be to factor the polynomials appearing in $f$; namely,
\begin{equation}
f(x) = \big(x-\big(n^2\sigma-n\sqrt{n^2\sigma^2+4}\big)\big)\big(x-\big(n^2\sigma+n\sqrt{n^2\sigma^2+4}\big)\big)e^x - \big(x-\big(-n^2\sigma-n\sqrt{n^2\sigma^2+4}\big)\big)\big(x-\big(-n^2\sigma+n\sqrt{n^2\sigma^2+4}\big)\big)e^{-3x}.
\end{equation}
Though, I don't see clearly how one may proceed from here.

There is a formal analytical solution to the equation.
Rewrite it as
$$e^{-4x}=\frac {x^2-\sigma n^2 x-n^2 } {x^2+\sigma n^2 x-n^2 }=\frac{(x-a)(x-b) }{(x-c)(x-d)}$$ and the solution is given in terms  of the generalized Lambert function (have a look at equation $(4)$ in the paper).
