4. 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) $(5,+\infty)$
(D)
4.【Analysis and Solution】From the given, we have $\frac{\sqrt{x^{2}+(y+1)^{2}}}{\left|\frac{x-2 y+3}{\sqrt{1^{2}+(-2)^{2}}}\right|}=\sqrt{\frac{5}{m}}$. This indicates that the ratio of the distance from $(x, y)$ to the fixed point $(0,-1)$ and the distance to the fixed line $x-2 y+3=0$ is a constant $\sqrt{\frac{5}{m}}$. By the definition of an ellipse, we get $\sqrt{\frac{5}{m}}5$.