Hypothesize is a is a hypothesis, which means a guess based off of limited evidence. A sentence starting with this could be "Hypothesize about this, the woman wasn't home at that time, so does that mean she could have committed the murder?" Or something along that line.
Hope I could help!
I would give you a better answer if I could see the paragraph, but my best guess is that productive means producing or able to produce large amounts of goods, crops, or other commodities. Also, achieving or producing a significant amount or result or producing or giving rise to.
