I've been trying to think of an intuitive way to understand Newtons Polynomial Interpolation method and boiled it down to solving pic related. I tried writing it out for a general number n, but it turned into a huge mess and I feel I'm missing something obvious. Any help is appreciated.
As for what the bottom section is showing, the nth difference of (x+?x)^n should give the rate of change the polynomial you're trying to interpolate through the given points y0,y1,y2 etc. More importantly, through you can use it to find the coefficient of your quadratic (n=2 in this case) term in the series. Though I do not believe that is not necessary for solving pic related.