paparts, I'm assuming this is a school assignment.
To start with this is called an "H-bridge", commonly used for motor direction control. Look at the possible input (A & B) states. For each input combination, use your transistor theory to see what happens to each of the four input transistors (Q5,6,7,8), and how these control each of the four output transistors (Q1,2,3,4), and thus the states of the outputs (Motor A and Motor B).
If it helps make four copies of the schematic and label the voltages at each point for each of the input combinations.
Is this cascaded? or cascoded? or should I solve it individually? All I know as of now is that they all act as a switch. This darlington transistors have high current gain and thats the problem I am about to findout. Is my idea correct? And also I'll divide it in to 2 since the other side is just the mirror of the other right?