What are the pros and cons of each of these methods? When might each method be most appropriate? Which method do you prefer? Explain why.

I agree with Paul H. Some methods are better in some situations that others.

Factoring is the easiest method if the equation lends itself to it.

If I find that factoring is not going to work, I often turn to the quadratic formula next.

I might use a graphing method to determine if the roots I calculated make sense, I'm not sure I would try and do it that way exclusively unless I had a graphical calculator handy. On the other hand, it is always useful to try and visualize a problem.

Completing the square is just a variation on using the quadratic formula, it takes a little more effort but can be a good way to get the roots.

Hope this helps!