What was the basis for russian claims to the northwest in the early 1800s? a) communities of settlers b) mining industry c) fur trade d) agriculture

I'd say the correct answer is C. fur trade.
As you already know, Russia is famous for its fur, especially given the temperatures in that country. Fur is important for Russians because they have to warm themselves one way or another. So they obviously wanted to expand into territories where they could continue that trade.

Democracy and darwinism would be my answer.