Learn Spanish (Latin America): Ella se agachó debajo del escritorio sin que el profesor la viera. - She crouched under the desk without the teacher seeing her.

The phrase sin que (without) always triggers the subjunctive. Since the main action se agachó (she crouched) is in the past tense, the following verb must be in the imperfect (past) subjunctive form viera.

Why couldn't we just use the infinitive here and say sin verla?

We use sin + infinitive only when the subject stays the same. If we said Ella se agachó... sin verla, it would mean she crouched without seeing someone else! Because the subject changes from the girl crouching to the teacher seeing, we must use sin que followed by a conjugated verb.

Why does the sentence use the reflexive se agachó instead of just agachó?

The non-reflexive verb agachar means to lower or bow a specific part of something, like agachar la cabeza (to lower one's head). When you lower your entire body to crouch or squat, you must use the reflexive form agacharse.

Why use la in la viera instead of le?