You die, and awake in Hell. Satan awaits you, and has prepared a curious game. On a blackboard, he has written the polynomial $x^2+x+666$. He explains the rule:
On each day at $12$ noon, you must either increase or decrease the coefficient of $x$ by $1$, and a minute after, Satan will either increase or decrease the constant term by $1$. If at some point, the polynomial on the board at that instant has integer roots, you'll be freed from Hell. Satan will of course try his hardest to make sure you never leave.
Is there a strategy that eventually guarantees your salvation? Or can Satan conspire to keep you in Hell forever?
Credits: Puzzle taken from Indian Maths Olympiad, 2014, wording copied from Coin Flipping Game with the Devil.