A problem by Shivam jha

Let's us write the expression as below : $\begin{aligned} & n^2 +20n +15 = x^2 \\& (n+10)^2 -x^2 = 85 \\& (n+x+10)(n-x+10) = 85\end{aligned}$ For $n +10 > x$ $(n+x+10)(n-x+10) = \begin{cases} & 85\times 1 \\& 5\times 17\end{cases}$ Equating the equations above and solving for $n$ we get $n= 33$ and $1$ .

For $n+10 < x$ $(n+x+10)(n-x+10) =\begin{cases} & -85\times -1 \\& -5\times -17\end{cases}$ Doing same as previous equations and solving for $n$ we get $n= -53$ and $-21$

Therefore, sum of all possible integers of $n$ is $1+33-53-21 = -20-20=-40$ .