Get it in the standard conic form:

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

then calculate the discriminant, which is B2 - 4AC

1. If the discriminant is positive, the equation represents a hyperbola.

2. If the discriminant is zero, the equation represents a parabola.

3. If the discriminant is negative then it is either an ellipse or a circle.
   If it's a circle, then B=0 and A=C, otherwise it's an ellipse.

So let's get

y2 + 2y = x2 + x

in the standard conic form 

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

Get 0 on the right by subtracting the right side from both sides:

y2 + 2y - x2 - x = 0

Now lets rearrange the terms in the order as they appear in the standard
conic form:

 Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

-1x2 + 0xy + 1y2 - 1x + 2y + 0 = 0
 
So A=-1, B=0, C=1, D=-1, E=2, F=0

B2 - 4AC = (0)^2 - 4(-1)(1) = 0 - (-4) = 0 + 4 = 4

That's positive, so the equation represents a hyperbola.

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Edwin