Answer
$${\text{ parabola}}$$
Work Step by Step
$$\eqalign{
& {x^2} + 2x = - 4y \cr
& {\text{Complete the square and factor}} \cr
& {x^2} + 2x + 1 = - 4y + 1 \cr
& {\left( {x + 1} \right)^2} = - 4\left( {y - \frac{1}{4}} \right) \cr
& {\text{The equation is written in the form }}{\left( {x - h} \right)^2} = 4p\left( {y - k} \right) \cr
& {\text{Therefore,}} \cr
& {\text{The graph of the equation }}{x^2} + 2x = - 4y{\text{ is a parabola}} \cr} $$
