<s>[INST] Find all integer solution Find all integer solutions such that $$a+1|2a^2+9$$
Solution. I could solve this by writing $$\frac{2a^2+9}{a+1}=2a-2+\frac{11}{a+1}.$$ So, the only integer solution for the last equation are $a=10, a=-12.$
But, i want to get a solution using divisibility properties.
 [/INST] By elementary means, since $a+1|2a^2+2a$, it divides $2a^2+9-(2a^2+2a)=9-2a$.
Since  $a+1|2a+2$, we have $a+1|(9-2a)+(2a+2)$ 
Hence $a+1|11$
Hence $a+1=±1$ or $a+1=±11$
Hence $a=-12,-2,0,10$.
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