In 1758, Lambert solved the trinomial equation $x = q+x^m$ by giving a series development for $x$ in powers of $q$ as stated in On the Lambert W function (2nd page).
Looking for the series solution of $x = q+x^m$ and the detailed process by which it was solved.
(I have done a lot of searching but couldn't find it anywhere.)
Let use solve $x=q+x^{n+1}$ using power series in $q$. The idea is to use an iterative process with $x_0=q+O(q^2)$ and $x_{k+1}=q+x_k^{n+1}$. In the limit we get $$x = q + q^{n+1}+(n+1)x^{2n+1}+(n+1)(3n+2)/2x^{3n+1}+\cdots$$ where the coefficients are OEIS sequence A070914. In general, you could use the Lagrange inversion theorem. See the example in the Wikipedia article which also mentions Lambert W.