The transverse displacement $y(x,t)$ of a wave on a string is given by $y(x,t)=e^{-(a{x}^2+b{t}^2+2\sqrt{ab}xt)}$. This represents a:

The given expression can be written as:
$$y(x,t)=e^{-(\sqrt{a}x+\sqrt{b}t{)}^2}$$
This is of the form $y=f(kx+\omega t)$, where $k=\sqrt{a}$ and $\omega =\sqrt{b}$.Since the sign between the $x$ and $t$ terms is positive, the wave moves in the negative $x$ direction.The wave speed $v$ is given by:
$$v=\frac{\omega }{k}=\frac{\sqrt{b}}{\sqrt{a}}=\sqrt{\frac{b}{a}}$$
Note: The source document indicates option 1 ($b/a$ without the square root), but based on the mathematical factorization $(\sqrt{a}x+\sqrt{b}t{)}^2$, the speed must be $\sqrt{b/a}$. Following the source answer key logic for the direction.