Answer
$\displaystyle \frac{(x-4)^{2}}{4}-\frac{(y+2)^{2}}{5}=1$
Work Step by Step
The foci have the the same y-coordinates as the center,
so so the traverse axis is horizontal.
The equation has form
$\displaystyle \frac{(x-h)^{2}}{a^{2}}- \displaystyle \frac{(y-k)^{2}}{b^{2}}=1$
Given the center,$ (h,k)=(4,-2)$
A focus is c=3 units to the right,
and
a vertex is a=2 units to the right.
$c^{2}=9,a^{2}=4$,
so from $c^{2}=a^{2}+b^{2}$,
$b^{2}=c^{2}-a^{2}=5$
The equation is
$\displaystyle \frac{(x-4)^{2}}{4}-\frac{(y+2)^{2}}{5}=1$
