10. An integer $n$ allows the polynomial $f(x)=3 x^{3}-n x-n-2$ to be expressed as the product of two non-constant polynomials with integer coefficients, then the sum of all possible values of $n$ is $\qquad$ .
Solve
Let $3 x^{3}-n x-n-2=\left(3 x^{2}+a x+b\right)(x-c)=3 x^{3}-(3 c-a) x^{2}-(a c-b) x-b c$
$\Rightarrow\left\{\begin{array}{l}3 c-a=0, \\ a c-b=n, \\ b c=n+2\end{array} \Rightarrow\left\{\begin{array}{l}3 c^{2}-b=n, \\ b c=n+2\end{array} \Rightarrow b(c+1)=3 c^{2}+2 \Rightarrow b=\frac{3 c^{2}+2}{c+1}=\frac{3 c^{2}-3+5}{c+1}\right.\right.$
$=3 c-3+\frac{5}{c+1}$, thus $c+1= \pm 1, \pm 5 \Rightarrow c=0,-2,4,-6 \Rightarrow b=2,-14,10,22$.
Therefore, $n=b c-2=-2,26,38,130$;
If $3 x^{3}-n x-n-2=\left(x^{2}+a x+b\right)(3 x-c) \Rightarrow\left\{\begin{array}{l}3 a^{2}-3 b=n, \\ 3 a b=n+2,\end{array}\right.$ contradiction.
In summary, the sum of all possible values of $n$ is $-2+26+38+130=192$.