The problem at the end of explanation #2 seems to overlook that no possible values of a are integers.
If you calculate each value of a, they come out to be 67/6, 35/6, 25/6, and 17/6. Their sum is 24, but none of them are integers. So there are no possible integer values that satisfy that equation.
The solution uses a trick to calculate the sum of all possible values without actually finding the values of a. So it overlooked the fact that there are no integer values of a.