In the following equation, capital letters represent arbitrary real numbers that are constant with respect to $x$:
$$A\left(x+B\right)\left(1 + \frac{C}{x+D}\right)^E + Fx + G = 0$$
I'm trying to solve for $x$; however, my algebra is failing me—especially when it comes to the exponential part.
Is it even possible to solve this equation? If so, how should one approach problems of this sort?
If $E$ is an integer, as MPW told, the equation will reduce to a polynomial which could be quite unpleasnt to solve if $E$ is large.
For the most general case, I think that the most efficient will be numerical methods for solving nonlinear equations $f(x)=0$, Newton being the simplest to use for a good efficiency (it only requires a "reasonable" starting estimate $x_0$ for each solution to be searched.
For illustration purposes (and since I am lazy), I used $A=1$,$B=2$,$C=3$,$D=4$,$E=5$,$F =6$,$G=7$ and I plotted the function. There are solutions close to $-5$ and $-2$.
So, let us search the first one using $x_0=-5$. Newton method updates the guess according to $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ So, for this case, the successive iterates will be $-5.10519$,$-5.17494$,$-5.19850$,$-5.20058$,$-5.20060$ which is the solution for six significant digits.
If we do the same using $x_0=-2$, the successive iterates will be $-1.95176$, $-1.94819$,$-1.94817$ which is the solution for six significant digits.