Answer
$\displaystyle \frac{x^{2}}{9}-\frac{y^{2}}{7}=1$
Work Step by Step
The foci have equal y-coordinates,
so the transverse axis is horizontal
and the equation is
$\displaystyle \frac{(x-h)^{2}}{a^{2}}-\frac{(y-k)^{2}}{b^{2}}=1$
The midpoint of the foci (the center):
$(\displaystyle \frac{-4+4}{2},\frac{0+0}{2})=(0,0)=(h,k)$.
The vertices are 3 units to the right/left
of the center, so $a=3.$
$(a^{2}=9)$
From the foci, $c=4$.
The remaining thing to find is $b^{2}$:
$b^{2}=c^{2}-a^{2}=16-9=7$
The equation is
$\displaystyle \frac{x^{2}}{9}-\frac{y^{2}}{7}=1$
