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On TV, someone with a grave expression is talking about the problems facing our country's future.
Literal
TV [on-では] our-country [genitive-の] future [genitive-の] problems [object-を] someone [subject-が] grave face [with-で] is-talking.
A media-watching observation. 深刻な顔 ('grave face,' 'serious expression') is a stock phrase for the expression of someone speaking about weighty matters. で here marks the manner of the speaking ('while making a grave face') — distinct from instrumental, locative, and causal で. しゃべる ('to talk, chat, speak') is a more casual/colloquial alternative to 話す, with a slight nuance of informality or continuous speech. The ~ている progressive form marks the observation as an ongoing scene the speaker is witnessing. わが国 ('our country') is the formal/elevated self-reference for one's nation.