Why does +bc become negative in the proof of the Euclidian Algorithm?

[MienTommy]

In this video, at 5:35 He has d/(a-qb) for the first part. I was not sure how he got that. Why is it not d/(a+qb)?

Because d/a and d/bc implies d/(a+bc)

Why does +bc become negative?

 

[mfb]

It divides both.

If a=x*d and b=y*d then a-qb=x*d - q*y*d = (x-qy)*d and also a+qb=x*d + q*y*d = (x+qy)*d
The second formula less useful for the next step, however, because it would make numbers larger.