Solve the equation $xy=x+2y+2009$ in integers

Question

I know that the left side is a hyperbola and the right hand side is a line. So they have at most 2 solutions. I set $xy=k$ and solved for $y$, and after that substituted it on the right side.

The numbers are getting too big and the corresponding quadratic equation seems to be unsolvable without a calculator. Thank you.

Answer 1

You can rewrite the equation as $(x-2)(y-1) = 2011.$

Answer 2

Just to finish up @md2perpe's solution:

$$\{ \{x = -2009, y = 0\}, \{x = 1, y = -2010 \}, \{x = 3, y = 2012\}, \{x = 2013, y = 2 \} \}$$