Find all integers of the form $\frac{(x-1)^2(x+2)}{2x+1}$ I made a python program and I have that
\begin{array}{|c|c|}
\hline x & \frac{(x-1)^2(x+2)}{2x+1} \\\hline
  -14 & 100 \\\hline
-5& 12 \\\hline
-2&0 \\\hline
-1&4 \\\hline
0&2 \\\hline
1&0 \\\hline
4&6 \\\hline
13&80 \\\hline
  \end{array}
And taking very large intervals of integers this seems like the only integers of this form.
However I don't know how can i proved.
Any ideas?

Note that $\gcd(x,2x+1)=1$, then:
$$\frac{(x-1)^2(x+2)}{2x+1}=\frac{x^3-3x+2}{2x+1}=\frac{x^3-7x}{2x+1}+2=\frac{x(x^2-7)}{2x+1}+2\iff \\
\frac{x^2-7}{2x+1}=\frac{x^2+14x}{2x+1}-7=\frac{x(x+14)}{2x+1}-7 \iff \\
\frac{x+14}{2x+1}=14-\frac{27x}{2x+1} \iff \\
\frac{27}{2x+1} \iff \\
x\in \{-14,-5,-2,-1,0,1,4,13\}$$
