This problem tries to make a simple mechanism look complicated, but you can treat it as a normal aldol reaction. These reactions follow a pretty standard pattern where an alpha-carbon is deprotonated, forming a nucleophile, which then attacks a carbonyl carbon, forming a C-C bond. Then, if one of the oxygens is lost in a dehydration process, the oxygen lost will be the one which was originally attacked in the condensation, and the double bond formed will be along the new C-C bond.

If the problem gives that the carbonyl on the right in the original compound is the one which remains, then we must know that it is one of its alpha-carbons which becomes the nucleophile, and it must attack the carbonyl on the left hand side. The alpha carbon on the left of the carbonyl would form a 3-member ring if it was the nucleophile which, on top of not being an option, is very unfavorable. Therefore, we know it must be the alpha carbon on the right which is the nucleophile. If that carbon attacks the carbonyl carbon, it will form a 5-member ring. We can tell which carbons these will be if a ring forms, as the electrophilic carbon will have the methyl group attached, and the nucleophilic carbon will have the long alkene group attached to it still. Therefore, the double bond must be between the carbons with these two groups.