[Solution] (1) When $m=0$, the equation becomes $-2(1+x)=0$, which has a solution, so (A) is not true;
(2) When $m \neq 0$, the equation can be transformed into
$$
\left(m^{2}-m-2\right) x+(m-2)=0 \text {, }
$$

If $m^{2}-m-2=0$, that is, $m=2$ or -1.
When $m=-1$, $m-2 \neq 0$, the equation has no solution, thus (B) is false;
When $m=2$, $m-2=0$, the equation has infinitely many solutions.
Therefore, the correct choice is $(C)$.
$2 \cdot 35$ There is an equation involving an unknown $x$
$$
m^{2} x+m(1-x)-2(1+x)=0,
$$

where $m$ is a constant. Among the following statements, the correct one is
(A) $m$ cannot be 0, otherwise the equation has no solution.
(B) Regardless of the value of $m$, the equation always has a unique solution.
(C) When $m$ is a certain positive number, the equation has infinitely many solutions.
(D) When $m$ is a certain negative number, the equation has no solution or infinitely many solutions.
(2nd "Jinyun Cup" Junior High School Mathematics Invitational Competition, 1985)