Answer
Vertex: $(0,0)$
Focus: $\left(-\frac{5}{12},0\right)$
Directrix: $x=\frac{5}{12}$
Work Step by Step
Recall: The parabola $y^2=4px$ has a vertex $(0,0)$, a focus $(p,0)$ and a directrix $x=-p$.
We have $5x+3y^2=0$ or equivalently $y^2=\frac{-5x}{3}$.
Then,
$4p=-\frac{5}{3}$
$p=-\frac{\frac{5}{3}}{4}$
$p=-\frac{5}{12}$
So, the vertex is $(0,0)$, the focus is $(-\frac{5}{12},0)$, and the directris is $x=\frac{5}{12}$.
Sketch the graph: