Example 1 (1997 National High School League Question) In the Cartesian coordinate system, if the equation $m\left(x^{2}+y^{2}+2 y+1\right)=(x-2 y+$ $3)^{2}$ represents an ellipse, then the range of values for $m$ is ( ).
A. $(0,1)$
B. $(1,+\infty)$
C. $(0,5)$
D. $(5,+\infty)$
Solution: Choose D. Reason: If we directly convert the ellipse to the standard equation $\frac{\left(x-x_{0}\right)^{2}}{a^{2}}+\frac{\left(y-y_{0}\right)^{2}}{b^{2}}=1$, the process would introduce an $xy$ term, which requires rotation. However, rotation is not covered in high school. Therefore, we can only use the definition of conic sections, through the eccentricity $e$ which satisfies $0<e<1$.