Problem:
Given the equation $-5 x^2+4 x+4 y^2+10 y+7=0$, rewrite and classify the conic and state relevant properties.
Answer:
Classification: Hyperbola
Equation: $4 \left(y+\frac{5}{4}\right)^2-5 \left(x-\frac{2}{5}\right)^2=-\frac{31}{20}$
Foci: $\left(
\begin{array}{cc}
 \frac{1}{20} \left(8-3 \sqrt{31}\right) & -\frac{5}{4} \\
 \frac{1}{20} \left(8+3 \sqrt{31}\right) & -\frac{5}{4} \\
\end{array}
\right)$
Eccentricity: $\frac{3}{2}$
Center: $\left\{\frac{1}{2} \left(\frac{1}{20} \left(8-3 \sqrt{31}\right)+\frac{1}{20} \left(8+3 \sqrt{31}\right)\right),-\frac{5}{4}\right\}$
Asymptotes: $\left\{y=\frac{\sqrt{5} x}{2}-\frac{1}{\sqrt{5}}-\frac{5}{4},y=\frac{1}{20} \left(4 \sqrt{5}-25\right)-\frac{\sqrt{5} x}{2}\right\}$
{'source': 'AMPS', 'file': 'amps/mathematica/algebra/conic_sections/13817.txt'}