Example 7 (2003 Anhui Province Competition Question) The number of intersection points of the curves $2 x^{2}-x y-y^{2}-x-2 y-1=0$ and $3 x^{2}-4 x y+y^{2}-3 x+y=0$ is ( ).
A. 2
B. 3
C. 4
D. Infinitely many
Solution: Choose D. Reason: By performing degeneration, from $2 x^{2}-x y-y^{2}-x-2 y-1=0$ we get $(2 x+y+1)(x-y-1)=0$, meaning this curve degenerates into two lines $2 x+y+1=0$ or $x-y-1=0$; from $3 x^{2}-4 x y+y^{2}-3 x+y=0$ we get $(3 x-y)(x-y-1)=0$, meaning this curve degenerates into two lines $3 x-y=0$ or $x-y-1=0$. Since one pair of lines from the degenerated curves coincide, the two curves have infinitely many common points.