Problem - 2854
Show that $(1 + x + \cdots + x^n)^2 - x^n$ is the product of two polynomials.
Let $S_k=1+x+\cdots + x^k$. Then $$\begin{align}&(1+x+\cdots + x^n)^2-x^n\\ = & (S_{n-1} + x^n)^2-x^n\\=&S_{n-1}^2 + 2S_{n-1}x^n + x^{2n} - x^n\\=&S_{n-1}^2 +2S_{n-1}x^n + (x^n-1)x^n\\=&S_{n-1}^2 +2S_{n-1}x^n + S_{n-1}(x-1)x^n\\=&S_{n-1}(S_{n-1}+2x^n+(x-1)x^n)\\=&S_{n-1}(S_{n-1}+x^n+x^{n+1})\\=&(1+x+\cdots + x^{n-1})(1+x+\cdots + x^n + x^{n+1})\end{align}$$