Answer
Foci: $(1, -3+\sqrt{3})$ and $(1, -3-\sqrt{3})$.
Work Step by Step
$\displaystyle \frac{(x-1)^{2}}{2}+\frac{(y+3)^{2}}{5}=1$
(major axis vertical),
$a^{2}=5,$
$ a=\sqrt{5}\approx$2.236
$b^{2}=2$,
$ b=\sqrt{2}\approx$1.414
Center: $(1, -3)$
$c^{2}=a^{2}-b^{2}=5-2=3$
$ c=\sqrt{3}\approx$1.732
Foci are $c$ units above and $c$ units below the center.
Foci: $(1, -3+\sqrt{3})$ and $(1, -3-\sqrt{3})$.