<s>[INST] Can someone show me why this factorization is true? $$x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + \dots + xy^{n-2} + y^{n-1})$$
Can someone perhaps even use long division to show me how this factorization works? I honestly don't see anyway to "memorize this". I like to see some basic intuition behind this
 [/INST] Cancelation:
$$
\begin{array}{cccccccccccccccc}
x & (x^3 & + & x^2 y & + & xy^2 & + & y^3) \\
  &      & -y & (x^3 & + & x^2 y & + & xy^2 & + & y^3) \\[25pt]
= & x^4 & + & x^3 y & + & x^2y^2 & + & xy^2 \\
  &     & - & x^3y & - & x^2y^2 & - & xy^2 & - & y^3 \\[25pt]
= & x^4 & & & & & & & - & y^4
\end{array}
$$
</s>