Hi an old question from the late Edwin Weiss's A First Course in Algebra and Number Theory:
Find all n such that $n(180+n)$ is a perfect square.
Prof Weiss, who wrote marvelous problems, passed 25 years ago, and his book, which we used in his course in 1979, is long out of print. But I would be grateful to have an answer here.
We have $$n^2+180n=k^2$$ implying $$n^2+180n+8100=k^2+8100$$
Hence $(n+90)^2-k^2=8100$ , hence $(n+90-k)(n+90+k)=8100$
Now determine all integer pairs $(a/b)$ with $ab=8100$ and solve the equation system $$n+90-k=a$$ $$n+90+k=b$$ If this system has a solution in integers, this solution also solves the given equation.