<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] Since you said above that you already believe that the result is true, that makes me think you really want to know how to come up with the result (if it had not been presented to you).
My answer would be: play with several examples! 
That is, sharpen your pencil and get busy factoring out $x^n-y^n$ for $n=1,2,3,\dots$ using long division or any other tools at your disposal. Then look at the pattern formed by the coefficients as well as the powers of $x$ and $y$:
\begin{array}{c|c}
 n & x^n-y^n \\\hline
 1 & x-y \\
 2 & (x-y) (x+y) \\
 3 & (x-y) \left(x^2+y x+y^2\right) \\
 4 & (x-y) \left(x^3+y x^2+y^2 x+y^3\right) \\
 5 & (x-y) \left(x^4+y x^3+y^2 x^2+y^3 x+y^4\right) \\
 6 & (x-y) \left(x^5+y x^4+y^2 x^3+y^3 x^2+y^4 x+y^5\right) \\
 7 & (x-y) \left(x^6+y x^5+y^2 x^4+y^3 x^3+y^4 x^2+y^5 x+y^6\right) \\
 8 & (x-y) \left(x^7+y x^6+y^2 x^5+y^3 x^4+y^4 x^3+y^5 x^2+y^6 x+y^7\right) \\
 9 & (x-y) \left(x^8+y x^7+y^2 x^6+y^3 x^5+y^4 x^4+y^5 x^3+y^6 x^2+y^7 x+y^8\right) \\
 10 & (x-y) \left(x^9+y x^8+y^2 x^7+y^3 x^6+y^4 x^5+y^5 x^4+y^6 x^3+y^7 x^2+y^8 x+y^9\right) \\
\end{array}
Hopefully this would lead you to conjecture that the formula you gave is indeed true for all $n$. Then to prove this, you would proceed by induction.
Hope that helps.
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