この問題をどうやって扱いましょうか。

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 do; to undertake; to perform; to play (a game); to study; to send; to dispatch; to despatch; to put; to move; to turn (one's head, glance, etc.); to give (esp. to someone of equal or lower status); to let have; to present; to bestow; to confer; to make (a vehicle) go faster; to run (a business); to keep; to be engaged in; to practice (law, medicine, etc.); to practise; to have (food, drink, etc.); to eat; to drink; to smoke; to hold (a performance); to perform; to show; to ease (one's mind); to harm; to injure; to kill; to have sex with; to live; to get by; to get along; to do ... completely; to do ... broadly; to do ... to a great distance; to do ... for (someone of equal or lower status); to do ... to (sometimes with negative nuance); to make active efforts to ...

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