Answer
Foci: $(4, \sqrt{21})$ and $(4, -\sqrt{21})$.
Work Step by Step
$\displaystyle \frac{(x-4)^{2}}{4}+\frac{(y-0)^{2}}{25}=1$
(major axis vertical),
$a^{2}=25,$
$a=5$
$b^{2}=4$,
$b=2$
Center: $(4, 0)$
$c^{2}=a^{2}-b^{2}=25-4=21,$
$c=\sqrt{21}\approx 4.583$
Foci are $c$ units above and $c$ units below the center.
Foci: $(4, \sqrt{21})$ and $(4, -\sqrt{21})$.