Methods for Finding Exact Solution For $e^{2x}+p(2x)$

Question

I know there are ways using the Lambert W function, and have had answers to simpler examples, for example $$e^{2x}+1+2x=0\Rightarrow e^{2x}=-2x-1$$ has the solution $$x=\frac{-1}{2}-\frac{1}{2}W\left(\frac{1}{e}\right)$$ which can be found on Wikipedia and is done through a transformation. What I can't seem to find is reference to higher degree polynomials. The equation above involves a linear equation. What about quadratics, cubics, and higher $n$-th degree polynomials? I know I can use Newton's method to find great approximations, but what about exact?

Answer 1

$$e^{2x}+1+2x=0\Rightarrow e^{2x}=-2x-1$$ $$\Rightarrow 2 \cdot x= -(e^{2x}+1)$$ $$(1) \quad \Rightarrow x= {{-(e^{2x}+1)} \over 2}$$ Using the theory of dynamical systems, we see that the right side of (1) has a derivative less than 1 around the values that I hope solve the above. In other words, there's a fair chance the next part will work. $$x= {{-(e^{2{{-(e^{2x}+1)} \over 2}}+1)} \over 2}$$ So, I just rewrote (1) in terms of itself. Keep going with this... $$x{{-(e^{2{{-(e^{2{{-(e^{2..}+1)} \over 2}}+1)} \over 2}}+1)} \over 2}$$

Using $-0.5$ as an estimate, I get an estimate of $-0.639...$