Question

7x^2 - 28x + 4y^2 - 8y= -2, hyperbola, ellipse, circle, or parabola?

Answer 1

http://en.wikipedia.org/wiki/Conic_section#Discriminant_classification
ellipse

Answer 2

After completing the squares, the equation becomes
\[7 (x - 2)^2 + 4 (y - 1)^2 = 30
\]
It is an ellipse of center (2,1) and with semi axes length
\[ \sqrt{\frac {30}{7}}\\
\sqrt{\frac {30}{4}}
\]