These mechanisms are similar to those in problem 738, only under basic conditions.
a) Carbonyl to Enolate (basic)
In the mechanism below the carbonyl was deprotonated at the alpha position, and the enolate was then drawn as a resonance form. You can also go straight from carbonyl to enolate in one reaction arrow. (rip off the proton, form the new double bond, and send the carbonyl "up" to form a negative charged oxygen).
b) Enolate to Carbonyl (basic) "DOWN"
Like a), this reaction can be done in one reaction arrow (negative charge comes "down" and double bond attacks a proton)
It's important that you become familiar with the mechanisms for enol and enolate formation in both acid (Q738) and in base (this problem), as well as carbonyl hydrate formation in both acid (Q706) and base (Q705).
If you can draw these four mechanisms you will be able to figure most of the reactions in second semester organic chemistry!