I just learned of extraneous solutions on the internet and thought, "could you both lose and gain solutions in one equation?" I think that, yes, you should be able to do that. However I haven't been able to come up with an example.
My question is this:
What could be a simple equation where the most straightforward way of solving it will result in you both gaining and losing a solution?
I'm not exactly sure what you mean, but sometimes it makes life easier to multiply an equation by $1$ and thus "both gain and lose a solution", like so: if $f$ is a polynomial (over $\mathbb{R}$, say), then consider $$\begin{align}f(x)&=\frac{x-a}{x-a}f(x)\\ &=1\cdot f(x) \\ &=f(x)\end{align}$$ for some $a\in\mathbb{R}$. It's difficult to manufacture an exact scenario where this would be optimal, though, although it does happen.
NB: Basically, the zero in the denominator is cancelled by the same zero in the numerator, so division by $0$ is avoided.
Also, as coolpapa points out in the comments, one should never really lose a solution when solving an equation properly (as far as I'm aware).