LEBANON, N. H.—While the rest of the Republican presidential field is betting on an electorate furious with President Barack Obama and angry with career politicians, John Kasich is making the opposite wager.

The Ohio governor is stumping across New Hampshire highlighting his experience in Congress, backing such issues as the Iran nuclear deal and campaign-finance reform that are usually supported by Democrats and painting himself as an uplifting alternative to the doom-and-gloom tone dominating the GOP primary.

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