I found this at a conference somewhere and it is the only way I've ever introduced factoring x^2 + bx + c.
It was called diamond math but I think it makes more sense to call it x-factor, mostly because it is a type of factoring.
I give this to students at the start of class and tell them I have a puzzle for them to figure out. I tell them the first two are done as examples for them and I want them to figure out the rest.
Then....we wait. There are always a few who figure it pretty quickly. I make them work 3-4 in a row to prove to me they know what's going on. I don't let students help each other or tell the answers.
Then there are a few who immediately start complaining...these are usually the students who get good grades but are not used to actually thinking.
After a few minutes/complaints I tell them to figure out a pattern with the numbers that can be repeated for every x.
This seems to help some. I let this drag out. I act very unhelpful. We wait some more. I try to hold out until every person has figured it out. If I can tell some people are getting verrrrry frustrated, I go to them one-on-one and try to prompt them with questions only.
From there, I give them a quadratic expression like x^2 + 6x + 8. I tell them that the c always goes on top and the b always goes on bottom and that we are always looking for the left and right numbers. I show them how to write the answer (x + 4)(x + 2) and then we practice.
A lot. A lot a lot.
Tip: Give students expressions with variables other than x to make sure they realize that the answer is written with the variable from the problem, not always an x.
See this post for my INB pages on x-factoring.