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?

Assuming $x$ is an integer
as $2x+1$ is odd
$2x+1$ will divide $(x-1)^2(x+2)$
iff $2x+1$ divides $(2x-2)^2(2x+4)=u$
Let $2x+1=y, u=(y-3)^2(y+3)\equiv27\pmod y$
So, the necessary & sufficient condition is: $y=2x+1$ must divide $27$
