ausstrecken
to stretch out
To straighten and extend a body part, such as an arm or a leg.
Er streckte die Hand aus, um sie zu begrüßen.
He stretched out his hand to greet her.

to stretch out
Loading your saved state.
to stretch out
To straighten and extend a body part, such as an arm or a leg.
Er streckte die Hand aus, um sie zu begrüßen.
He stretched out his hand to greet her.

'Ausstrecken' (to stretch out) is a separable verb. The prefix 'aus-' comes at the end of the sentence in a main clause. In subordinate clauses or in the infinitive with 'zu', the verb stays together.
Sie streckte die Beine unter dem Tisch aus.
She stretched her legs out under the table.
In its literal meaning, 'ausstrecken' (to stretch out) is almost always used with body parts. Common combinations are 'die Arme ausstrecken' (to stretch out one's arms), 'die Beine ausstrecken' (to stretch out one's legs), or 'die Hand ausstrecken' (to stretch out one's hand).
Das Kind streckte die Arme nach seiner Mutter aus.
The child stretched out its arms toward its mother.
The reflexive form 'sich ausstrecken' (to stretch out) means to lie down and make your body long in order to relax. The reflexive pronoun is crucial for the meaning here.
Die Katze streckte sich genüsslich in der Sonne aus.
The cat stretched out contentedly in the sun.
The idiom 'die Fühler ausstrecken' (to put out feelers) is a fixed expression. It is often used with the preposition 'nach' to name the target of the exploration.
Vor der Investition streckte die Firma ihre Fühler nach potenziellen Partnern aus.
Before the investment, the company put out feelers for potential partners.
to explore / investigate
'erkunden' and 'die Fühler ausstrecken' (to put out feelers) can both mean gathering information or checking possibilities. 'erkunden' is more direct and neutral. You can use it to say that someone wants to find something out by asking questions or investigating. 'die Fühler ausstrecken' is figurative and sounds more cautious. You are not deciding anything yet, but first seeing what opportunities there are. If it is about a clear investigation, 'erkunden' fits better. If someone is cautiously checking contacts or possibilities, 'die Fühler ausstrecken' is more natural.

Bei der Arbeit strecke ich schon die Fühler nach einer neuen Stelle aus.
At work, I am already putting out feelers for a new job.
Samir erkundet vor dem Umzug, welche Schulen es im neuen Stadtteil gibt.
Before moving, Samir investigates which schools there are in the new neighborhood.