This is the 15447th most frequent Russian word.
предлагая
"Предлагая" means "offering" or "proposing" (present participle).
In this sentence, 'предлагая' (offering) is used as a present participle to describe an ongoing action occurring simultaneously with the main action of the sentence.
Он подошёл ко мне, предлагая помощь.
He approached me, offering help.
Here, 'предлагая' (offering) expresses a simultaneous action performed by the subject, enriching the description of the main verb 'улыбнулась' (smiled).
Она улыбнулась, предлагая своё решение проблемы.
She smiled while offering her solution to the problem.
In this example, 'предлагая' introduces an explanation or cause, indicating the reason associated with the subject's hope for support.