The graph (x + 4)^2 - (y - 8)^2 = 39 is the graph of a(an) _______.

Eclipse or hyperbola?

first an ellipse formula is

(x-h)^2/a^2 + (y-k)^2/b^2 = 1
so this is NOT an ellipse

If that + were a - it would be a hyperbola
So
This is a hyperbola.