Is this a new formula? If so, what is it called?

Both!

It is a known formula -- it's the only possible product when the complex conjugate root factors are multiplied together -- but you did discover it on your own, and you get credit for it!

I spend a lot of time tutoring students and do my best to help them discover things just like this -- because it has a bigger impact on them (and thus, is easier to remember or to be able to derive again at a later date). The educational journey is much stronger this way than if I were to just stand up in front of them and rattle of formulas and rules and laws for them to memorize.

Keep up the good work, John!

Thanks for the reply! It is definitely a formula that's stuck in my head! I was pretty excited to figure it out as it speeds the process up tremendously!

Thanks again for the reply Doug!

Careful with the arithmetic, though -- your function (by using your formula!) should actually be 

f(x) = x^2 - 10x + 50

(and your formula has an extra parenthesis you don't need after the equal sign, but that's just a typo)

Check out more detail (and an alternate form of your formula) here: 

https://www.wolframalpha.com/input/?i=y%3D(x-a-b*i)(x-a%2Bb*i)