This is the 8205th most frequent Japanese word.
差し出す
"To hold out, present, or submit."
Here, '差し出す' is used to describe the act of physically extending or offering something, in this case, a hand.
彼は助けを求めて手を差し出した。
He extended his hand seeking help.
In this sentence, '差し出す' means to present or offer something formally, such as documents.
彼女は上司に書類を差し出した。
She handed the documents to the boss.
Here, '差し出す' is used metaphorically, implying offering solutions or suggestions in the context of a discussion.