Sherlock Holmes: "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth."
The question is not a mathematical one, it is more an anti-mathematical one, but the idea is kinda cute. Of course as a Differential Equations question, after a little examination it makes no sense. Let us solve it in almost Sherlock Holmes style. (Sherlock did not know much about singularities. Moriarty would not be impressed.)
Put $y=-1$, and substitute the various suggested values of $y'$ into the equation. We end up with four quadratic equations in $x$. Two of them, corresponding to the suggested answers $4/3$ and $16/5$, have no real solution. The one that comes from the suggested answer $-16/5$ gives $x=1/4$ or $x=-5/4$. The $x=1/4$ is discarded because we were told that there $y=1$. The $x=-5/4$ is presumably to be discarded, for reasons mysterious to me. Maybe there is a glimmer of a genuine differential equations reason, since we cannot get information about the situation when $x<0$ from the initial condition (but of course we cannot get information about $y<0$ either).
So what remains, namely (B), must be the truth. We all know that the argument that led to choosing (B) makes no sense. Someone might want to fool around with the numbers and produce a similar joke question that does not bump into singularity issues. Or not.