Let $q\in \prod_{n-1}$ be interpolation polynomial for $x_1,...,x_n$ and $p\in \prod_{n-1}$ be interpolation polynomial for $x_0,...,x_{n-1}$. Show that function $$N(x)=q(x)\left( 1+\frac{x_n-x}{x_0-x_n}\right) - \frac{x_n-x}{x_0-x_n}p(x)$$ is interpolation polynomial for $x_0,...,x_n$.
My idea was to represent $p(x)$ and $q(x)$ as Lagrange polynomials. Hence $$q(x)=\sum_{k=1}^nf(x_k)\prod_{i=1 \\ i \neq k}^n\frac{x-x_i}{x_k-x_i}$$ $$p(x)=\sum_{k=0}^{n-1}f(x_k)\prod_{i=0 \\ i \neq k}^{n-1}\frac{x-x_i}{x_k-x_i}$$ and just write it in $$N(x)=q(x) + \frac{x_n-x}{x_0-x_n}(q(x)-p(x))$$ but I'm stuck and feel like this not the right way to do it. Any ideas?
I found this although I'm curious if one can do this with Lagrange.
I think you are making this harder than it needs to be.
What you need to show, if I am interpreting your statements correctly, is that $N(x) =p(x) $ for $x=x_0,...,x_{n-1}$ and $N(x) =q(x) $ for $x=x_1,...,x_{n}$.
If $p$ and $q$ interpolate to the same values, you need to have $p(x) =q(x) $ for $x=x_1,...,x_{n-1}$.
This should be enough.