Why they used (stating) not( state)...althought they was talking about fact?
, the most comprehensive effort at measuring public opinion across more than 100 countries, the proportion of respondents (stating) that “having a democratic political system” is a “bad” or “very bad” way to run the country,

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Because the verb in the quote from the respondents is in the infinitive, so the verb "state" is in the infinitive also. Sorry, English sucks.

Evden çıkmadan dil öğrenme fırsatını kaçırmayın. Deneyimli dil eğitmenlerimizden oluşan seçkimize göz atın ve ilk dersinize şimdi kaydolun!