jmc

The left-hand side contains $x^2$ and $y^2$ terms with opposite signs. But be careful! The right-hand side, when expanded, contains the term $-8y^2,$ and so when all the terms are moved to the left-hand side, the terms $x^2$ and $4y^2$ will appear. Because the coefficients of $x^2$ and $y^2$ are of the same signs but unequal, this conic section is an $\boxed{(\text{E})}$ ellipse.