How do you solve this part of the equation?

Thank you for your responses. This makes a lot more sense now and has filled the gap in logic that I was missing. Thanks again!

In general, if you need to factor a quadratic, here's the safest way to think about it.

1. Put the quadratic in order, making sure that the coefficient of x^2 is 1: x^2 + b x + c = 0.

2. Look at the middle coefficient (b) and the last coefficient (c).

3. Find two numbers that multiply together to create c, and add together to create b.

4. Those two numbers represent the two question marks in the factored quadratic: (x + ?)(x + ?) = 0.

So, if we use the example you gave:

(x^2 - 2x + 1)

You'll look for two numbers that multiply to 1, and add to -2. Those numbers are -1 and -1. So, the factored version is (x - 1)(x - 1).

For another example, try the same with the following quadratic:

x^2 - 5x + 6

The two numbers should multiply to 6, and add to -5. The numbers are -2 and -3. So, the factored version is (x - 2)(x - 3).

If you have time, finish by double-checking your work with FOIL: multiply out the quadratic and make sure it matches what you started with.

I am often struggling with the highlighted steps in particular ...especially when it gets to really high numbers.

Does anybody have some tips / shortcuts to quickly find "two numbers that multiply together to create c, and add together to create b" ? Or do I just need to test numbers in my head until they work? These steps always consume most of my time for these questions..

Appreciate any help!

Not everybody finds this intuitive! The reason I usually teach it that way is because most people find that strategy more intuitive than the other way of doing it. But in your case, it might be the opposite, and that's completely fine.

The other option, which I encourage you to try, is to memorize and use the quadratic formula: https://www.purplemath.com/modules/quadform.htm Try some problems both ways, and stick with the one that's faster.

If you find that you need to just keep practicing the method I described in my last post, instead, my advice is to try factoring C before you do anything else. Once you have its prime factorization, it should be pretty obvious which pairs of numbers (or at least, which integers) could work - then check them to see if they add to B. It helps to have a sense of about how large the numbers ought to be, too... if B is 1 (for instance) you're probably looking for a positive and negative that are very close to each other, so you'll try to factor C into two very close values.