you are correct it will be negative for every X.
so its a maximum point
so the derivative before point x=1/a is positive
ans the derivative after point x=1/a in negative.
by i cant see where we got two negative point with a poisitive point in the middle
then we have two solution
what if it equals 0??
there is no place for being between negative and positive
how do you say that there is 1 solution in that case
its not -infinity like you said