Can you explain how you came up with the chart? I can track you all the way up to there. After you came up with the inverses, I see how you plugged in the numbers with a and a inverse, but I don't see how you formed the right hand side statements (which yielded +-1 roots).
So when you are referring to finding 3 roots for "both" cases, I have this so far:
Case 1: (x^2)+2x+1 mod 9 ? roots = ? without the mod 9 I have (x+1)(x+1)...
Case 2: (x^2)-2x+1 mod 9 ? roots = ? without the mod 9 I have (x-1)(x-1)...
So I have 2 roots for each case of +1 -1 which is what I started with... Where am I going wrong?