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.

You pretty much have the right idea. Observe that $a+1 | 2(a^2-1)$. Moreover we can write 
$$2a^2+9=2(a^2-1)+11.$$
Therefore for $a+1$ to divide $2a^2+9$, it has to divide $11$ (because $11$ is the difference of two expression both of which are divisible by $a+1$).
Thus the only values of $a+1$ are all divisors of $11$, namely $\pm 1, \pm11$. Thus $4$ solutions in all.
