Complete the Square to Determine Whether the Equation $x ^ { 2 } + 4 x - 8 y + 36 = 0$

Essay

Complete the square to determine whether the equation $x ^ { 2 } + 4 x - 8 y + 36 = 0$ represents an ellipse, a parabola, a hyperbola, or a degenerate conic. Then sketch the graph of the equation. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. If the equation has no graph, explain why.