4. In the Cartesian coordinate system, the curve represented by the equation $m\left(x^{2}+y^{2}+2 y\right.$ $+1)=(x-2 y+3)^{2}$ is an ellipse, then the range of $m$ is .
(A) $(0,1)$
(B) $(1,+\infty)$
(C) $(0,5)$
(D) $(5,+\infty)$
(Hunan contribution)
4. D.

From the given, $\frac{\sqrt{x^{2}+(y+1)^{2}}}{\frac{x-2 y+3}{\sqrt{1^{2}+(-2)^{2}}}}=\sqrt{\frac{5}{m}}$. This indicates that the ratio of the distance from $(x, y)$ to the fixed point $(0, -1)$ to the distance from $(x, y)$ to the fixed line $x-2 y+3=0$ is a constant $\sqrt{\frac{5}{m}}$. By the definition of an ellipse, $\sqrt{\frac{5}{m}}5$.