if $a^3+b^3 +3(a^2+b^2) - 700(a+b)^2 = 0$ then find the sum of all possible values of $a+b$ 
If $a+b$ is an positive integer and $a\ge b$ and
  $a^3+b^3 +3(a^2+b^2) - 700(a+b)^2 = 0$ then find the sum of all possible values of $a+b$.

I tried a lot to solve it, i came to a step after which i was not able to proceed forward, the step was $(a+b)^3 - 41\cdot17 (a+b)^2 - 3ab(a+b) - 6ab = 0$
Please help me to solve this.

Here is a start, not a full answer.
Let $s=a+b$ and $p=ab$. Then $3 p (s + 2) = (s - 697) s^2$ as WA tells us.
Then $s+2$ divides $(s - 697) s^2$ and so $s+2$ divides $2796 = 2^2 \cdot 3 \cdot 233$.
