Question 1141463
<br>
{{{x^2+x = y^2-2y}}}
{{{x(x+1) = y(y-2)}}}
{{{x(x+1)-y(y-2) = 0}}}<br>
The equation is satisfied when x(x+1) and y(y-2) are both 0:<br>
(1) x=0 and y=0;
(2) x=0 and y=2;
(3) x=-1 and y=0; and
(4) x=-1 and y=2<br>
Solutions: (0,0), (0,2), (-1,0), (-1,2)<br>
That does not preclude the possibility of other solutions; it seems unlikely that there would be.<br>
Entering "integer solutions to x^2+x=y^2-2y on wolframalpha.com shows only those four solutions.<br>