The first 4 you factored correctly.
Of the ones you made up, the first is correct, but the last 3 are not correct, even assuming the middle term has an as a factor.
As far as which order you write the factors, this does not matter, as per the commutative property of multiplication, i.e .
You are right about the last one, it cannot be factored in the traditional sense. In fact, it has complex roots. One way to check a quadratic to see if it can be factored is to see if the discriminant is a perfect square.
Assuming you are trying to solve for when a quadratic equals zero, once you have the quadratic factored, you then equate each factor to zero and solve for to find the roots, or values that make the quadratic zero.
If you have the product of some factors being equal to zero, then if any one of the factors is equal to zero, then the entire product is equal to zero. This is called the zero-factor property. If then we know either or must be true.