14. The set $M$ contains all the integral values of $m$ such that the polynomial
$$
2(m-1) x^{2}-\left(m^{2}-m+12\right) x+6 m
$$
has either one repeated or two distinct integral roots. Find the number of elements of $M$.
14. Answer: 4
If $m=1$, the polynomial reduces to $-12 x+6=0$ which has no integral roots.
For $m \neq 1$, the polynomial factorizes as $\left((2 x-m)((m-1) x-6)\right.$, with roots $x=\frac{m}{2}$ and $x=\frac{6}{m-1}$. For integral roots, $m$ must be even and $m-1$ must divide 6. The only possible values are $m=-2,0,2$ and 4 . So $M$ has 4 elements.