The best way to do these exercises is simply to read and understand each premise, as well as the conclusion. If you understand the statements, you will intuitively find how to prove the conclusion. Let's take a look at the following:
- EITHER (A AND B) OR (C AND D)
From the two premises, we know that A is the case (line 1), and we know that either both A and B are the case, or that both C and D are the case (line 2). When you have an EITHER-OR statement, you know that one or the other sides of the OR is the case (and possibly both). This means that if one of the two sides is not the case, the other side must be the case. This allows us to derive some interesting logical entailments. So, now that we understand the meaning of the two premises, let us look at the conclusion we are asked to derive.
Derive: IF NOT-(C AND D), THEN (A AND B)
This says that if C AND D is not the case, then A AND B is the case.
Well, yes, of course! This follows immediately from premise 2. Indeed, when you have an EITHER-OR statement (i.e. a disjunction), by definition one of the two sides must be true. Therefore, if one is false ('C AND D' in this exercise), then the other must be true ('A AND B' here).
So the answer to this exercise is simply to derive the conclusion from line 2, a step you must justify by appeal to the disjunctive syllogism (or disjunction elimination) rule (depending on the textbook you are using in class), applied to line 2.
By the way, premise 1 is useless to derive the desired conclusion. It's just there to mess with your head!