I'm a calculus 3 student and the bane of my existence has ALWAYS been factoring. It's just something I constantly struggle with and never learned formally. I'm always BLINDLY taking guesses and "brute forcing" factors with my calculator (which can take well over 25-50 tries easily, and most of the time I never find the answer).

Is there any way of doing it systematically? Does anyone know a good method? (I'm a calc 3 student, so I don't own an algebra book)

I can look at any polynomial like

which factors to

But I cannot see why! It looks like magic to me.

Ok... So I know simple factors like:

Which comes down to setting up

And finding something that multiplies to 4 and adds to 3. Those are easy. But honestly, those are about the ONLY ones I can do, and I'm coming across harder ones in my book... Although, I rarely see any that go above a cubic polynomial

Any advice/help would be appreciated. Thanks.