この問題をどう扱いましょうか。

this (something or someone close to the speaker (including the speaker), or ideas expressed by the speaker)

question (e.g. on a test); problem; problem (e.g. societal, political); question; issue; subject (e.g. of research); case; matter; question (i.e. doubt); public discussion; controversy; trouble; problem; inconvenience; difficulty

indicates direct object of action; indicates subject of causative expression; indicates an area traversed; indicates time (period) over which action takes place; indicates point of departure or separation of action; indicates object of desire, like, hate, etc.

how; in what way; how about

to deal with (a person); to treat; to handle; to take care of; to entertain; to deal with (a problem); to handle; to manage; to operate (e.g. a machine); to handle; to work; to deal in; to sell; to cover (a topic); to treat; to discuss; to take up; to treat A as B; to mediate (an argument); to be too much for one; to find unmanageable; to gossip

indicates a question (sometimes rhetorical); or; whether or not; some- (e.g. something, someone); indicates doubt, uncertainty, etc.; emphatic prefix; suffix forming adjectives or adverbs; in that way