This looks like a lesson in reduction using identities and such. I notice an important reduction available in each example. The left hand figure includes the function A'+A'B. If you think about this for a bit, you may realize that this is equal to A'. Consider that if A=1, then A'=0 and the term A'B must also equal 0 (no matter what B is, a 0 ANDed with it will always be 0). If A=0, then A'=1 and the first term insures that the overall result will be 1. So basically, it doesn't matter what B is at all. So the overall equation can be reduced from A'+A'B+AC = A'+AC.
In the right hand example, I notice that we have a term DBD'. When you AND something with the inverse of itself, the result must always be 0. When you include a 0 term in an OR, it does nothing, so you can simply omit it. So the second equation reduces to (A'B+A'CD)'.
Both of these results are easily realized using NAND gates, since a NAND is simply an OR using negative logic at the input.