Find all integers $(x,y)$ satisfying $(x+y+11)^2 = x^2 + y^2 + 11^2$ Find all integers $(x,y)$ satisfying $(x+y+11)^2 = x^2 + y^2 + 11^2$.
So far,
\begin{align*}
x^2 + y^2 + 121 + 2xy + 22x + 22y &= x^2 + y^2 + 121\\
2xy + 22x + 22y &= 0\\
(2x+22)y &= -22x\\
(x+11)y &= -11x
\end{align*}
At least 1 of $x,y$ must be a multiple of 11?
Dont know where to progress after this. All help appreciated.

Hint: First, dividing your second line by $2$ gives
$$xy + 11x + 11y = 0 \tag{1}\label{eq1A}$$
Next, using Simon's favorite factoring trick gives
$$(x + 11)(y + 11) = xy + 11x + 11y + 121 \tag{2}\label{eq2A}$$
