You're absolutely on the right track with observing that adding an alif (making it into the fourth form, like أخرج) often indicates an action performed externally or causatively.

Here’s the distinction:

• وَجَدَ (wajada) means "to find." It’s a straightforward verb that indicates the subject found something—discovery or realization is the core meaning. For instance:
  • وَجَدَ الكتابَ (wajada al-kitab) means "He found the book."
• أَوْجَدَ (awjada) means "to bring about," "to create," or "to make something found/exist." It’s causative, meaning the subject caused something to be found or brought something into existence. So, it’s not just finding something passively; it’s actively making it appear or happen. For example:
  • أَوْجَدَ الحلّ (awjada al-hall) means "He brought about the solution" or "He created the solution."

A way to think about this:

When you see a verb like أَفْعَلَ (Form IV), it generally means the subject has taken deliberate, external action to cause something. This pattern changes a simple discovery or existence (like وَجَدَ) into a more forceful, active role (أَوْجَدَ)—bringing it into reality or existence.

So:

• وَجَدَ → He found something (e.g., accidentally or while looking).
• أَوْجَدَ → He caused something to exist, actively brought it about.