Without the "xy" term it is particularly easy to tell:
If there is either or but not both, the graph is a parabola.
If the coefficients of and (when they are on the same side of the equation) are of opposites sign, it is hyperbola.
If the coefficients of and (when they are on the same side of the equation) are of the same sign, it is an ellipse.
Special case: if the coefficients are exactly the same, then the graph is a circle.
In this case...move y^2 to the other side
Divide both sides by ten to analyze the coefficients
Since the coefficients are equal and only one of the variable terms is quadratic, this is a parabola.
B = -4
C = -20
Probably this is in an upcoming chapter for you, but the xy term can be "rotated out" of the equation using a substitution. Doesn't matter, though. The catches them every time. I guess you could say it helps you 'discriminate' between the types ... which is why it's called the 'discriminant.'