factoring after finding roots with quadratic

I'm at uni now and I dont know why this is causing me problems or why I've only just discovered this problem.

Say you have a quadratic that you want to factorise. You solve it using the quadratic equation and you have the roots and then you can find the factors.

But the quadratic formula would give you the same roots for any multiple of the quadratic. Is the best way to sort this out just to expand it and work out what the difference is between the coefficients and then just put that multiple out the front of the factors?

Thanks,

Adam

(using brothers account)