TITLE: Simplifying $a(a-2) = b(b+2)$
QUESTION [5 upvotes]: I reduced a number theory problem to finding all ordered pairs $(a,b)$ that satisfy the equation $a(a-2) = b(b+2)$ in a certain range. After thinking about this for a while, I figured that either $a = b + 2$, $a = -b$, $a = b = 0$ or $a = 2 \text{ and } b = -2$. It is easy to prove that these values for $a$ and $b$ will all satisfy the equation, but how would I solve this equation had I not come up with these solutions? And how do I know I did not miss one? In short, can this equation be solved (more) rigorously?
Edit:
Silly of me I did not recognize that my last two solutions were already covered by my first two.

REPLY [2 votes]: $a(a - 2) = b(b + 2)$
$a^2 - 2a = b^2 + 2b$
$a^2 - b^2 = 2(b + a)$
$(a + b)(a - b) = 2(a + b)$
$(a + b)(a - b - 2) = 0$