On line 31 of the Missouri MO-1040, you should see the Resident Credit - this includes any taxes you paid to other states.
This is how states usually try to avoid double taxation on the same income.
However, this method is not perfect, because the states may levy income taxes at different rates, so the non-resident state's tax may not cover the tax of your resident state - that can't be helped.
Of course, SteamTrain's advice is solid - we can't be sure of anything until both states are released in TurboTax.
@SteamTrain
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