This is the 13389th most frequent Russian word.
надлежащую
Proper; appropriate.
Here, 'надлежащей' is used as an adjective in instrumental case modifying 'тщательностью', indicating the quality of carefulness as being appropriate or proper.
Она выполняет свою работу с надлежащей тщательностью.
She performs her work with due diligence.
In this sentence, 'надлежащие' is an adjective in nominative plural form modifying 'меры', showing that the measures should be adequate and proper for the situation.
Необходимо принять надлежащие меры для предотвращения происшествий.
It is necessary to take appropriate measures to prevent incidents.
Here, 'надлежащим' is used as an adverbial phrase modifier, in instrumental singular form integrated with 'образом', emphasizing the manner in which the signing should occur being proper and correct.