Answer
The vertices are $(3\pm7,-1)$.
The foci are $(3\pm \sqrt{65},-1)$.
Work Step by Step
The equation $$
\left(\frac{x-3}{7}\right)^{2}-\left(\frac{y+1}{4}\right)^{2}=1
$$
is hyperbola with center $(3,-1)$. We have $a=7$, $b=4$, $c=\sqrt{a^2+b^2}=\sqrt{65}$ and hence:
When the center is at the origin, the foci are $(\pm \sqrt{65},0)$ and the vertices are $(\pm 7,0).$ Then the translation by $(3,-1)$ gives
The new vertices are $(3\pm7,-1)$.
The new foci are $(3\pm \sqrt{65},-1)$.
