Question 1
Quadratic solitaire is a single-player game. To start the game, the player chooses two distinct nonzero integers $a$ and $b$ and writes the equation $x^{2}+ax+b=0$ on a blackboard. On each turn, if the equation currently written on the blackboard has two distinct nonzero integer solutions $x=u$ and $x=v$, then the player erases the equation and replaces it by one of the equations $x^{2}+ux+v=0$ or $x^{2}+vx+u=0$ of their choosing. Otherwise, the game ends.
Determine all initial choices of a and b such that quadratic solitaire can be played forever.
Determine all initial choices of a and b such that quadratic solitaire can be played forever.