When there is no coefficient on x^2 then it's easier (ie: a=1). Let p and q be the numbers we need to find. Because of FOIL (the method you used to expand) the two last terms need to multiply to make c and add to make b:
pq = c
p+q = b
c and b are given in the quadratic to be factorised and so you can solve for p and q, with practice it gets easier especially since p and q must be factors of c.
It is also possible that some do not factorise into rational numbers, you can check if your equation is one by checking the discriminant. If b^2-4ac (the discriminant) is a perfect square it can be factored rationally