I am looking at the following function on the domain $x\geq 0$:

$F(x)=(x+a)e^{x^2}(1-erf(x))-\frac{b}{\sqrt\pi}$,

where $a>0$, $0<b<1$ are parameters. From plotting this function for different values of $a$ and $b$ it seems that there is at most one root on $x\in[0,\infty)$. But how to prove it?