Answer
$\frac{(x-2)^{2}}{9} - \frac{(y+1)^{2}}{27} = 1$
Work Step by Step
General Equation of a Hyperbola is: $\frac{(x-h)^{2}}{a^{2}} - \frac{(y-k)^{2}}{b^{2}} = 1$
$c^{2} = a^{2} + b^{2}$
Vertices are at (-1, -1) and (5, -1), so the midpoint is the center, which is at (2, -1)
The vertices is on the horizontal major axis, so $a^{2}$ is under the x term
Center is (2, -1), thus h = 2, p = -1
The distance between center and vertice (-1, -1) is 3, thus a = 3 under the y term (since it's vertical)
Foci distance from center to (-4, -1) is 6, thus c = 6
$c^{2} = 7^{2} + b^{2}$
$6^{2} = 3^{2} + b^{2}$
$27 = b^{2}$
Thus the equation is $\frac{(x-2)^{2}}{9} - \frac{(y+1)^{2}}{27} = 1$
