So, we have taken
a, b, c & D and have considered all the possibilities of their signs, drawn a small graph for each. This is the first part with 8 cases for a>0 and we have another 8 cases below for a<0 i.e. inverted U graphs.
Do you see that 2 out of the 8 cases HAVE NO ROOTS
and strangely 2 out of these 8 cases are NOT POSSIBLE