Answer:
Our dog and our neighbor's dog keep barking.
Step-by-step explanation:
A compound subject is when two or more subjects share a verb or verb phrase. They are joined by conjunctions such as "and", "or", "nor" etc.
In the given sentences, the subjects "our dog" and "our neighbor's dog" share the same verb "bark". So, they can be merged into a compound subject by using the conjunction "and". This also makes the helping verb "keeps" plural, as the compound subject is plural.
Thus, the final sentence will be
Our dog and our neighbor's dog keep barking.