Consider the following equality
Is there any way to find the solutions of it in the closed form. Also, can we say anything about the number of solutions without solving it?
$$ (x+1)e^{\frac{(x+1)^2}{2a}}+(x-1)e^{\frac{(x-1)^2}{2a}}=4xe^{\frac{x^2}{a}} $$ where we know that $x=0$ is one of the zeros.
It is unlikely that there are closed-form solutions in general other than $0$. If you're interested in real solutions, it looks to me like there are three if $a<0$ or $0 <a < 1/(2 W(e^{1/2})-1)$ where $W$ is the Lambert W function ($1/(2 W(e^{1/2})-1)$ being the value of $a$ where your function has derivative $0$ at $0$).
I say "in general" because you may be able to solve for $a$ that makes a given $x$ a solution. If $x$ is rational, you get a rational function of $\exp(1/a)$.