The equation is
$\displaystyle
x^2-4xy+5y^2+2y-4=0$
What is the total number of pair of integers (x,y) that satisfy the above equation?
How do I solve it?
$\displaystyle \left\{\begin{array}{ll}(x-2y)^2=1\\(y+1)^2=4\end{array}\right.$
From here you have to solve four systems:
$\displaystyle \left\{\begin{array}{ll}x-2y=1\\y+1=2\end{array}\right., \ \left\{\begin{array}{ll}x-2y=-1\\y+1=2\end{array}\right., \ \left\{\begin{array}{ll}x-2y=1\\y+1=-2\end{array}\right., \ \left\{\begin{array}{ll}x-2y=-1\\y+1=-2\end{array}\right.$
and to keep the integer solutions.
Similarly for $\displaystyle \left\{\begin{array}{ll}(x-2y)^2=4\\(y+1)^2=1\end{array}\right.$