<s>[INST] Showing that $ 1 + 2 x + 3 x^2 + 4 x^3 + \cdots + x^{10} = (1 + x + x^2 + x^3 + x^4 + x^5)^2$ I was studying a polynomial and Wolfram|Alpha had the following alternate form:
$$P(x) = 1 + 2 x + 3 x^2 + 4 x^3 + 5 x^4 + 6 x^5 + 5 x^6 + 4 x^7 + 3 x^8 + 2 x^9 + x^{10} = (1 + x + x^2 + x^3 + x^4 + x^5)^2$$
Of course, we can verify this through expansion, but if I were a mathematician without access to CAS, how might I notice that this is the case?
I suppose what I'm asking is how one should "see" that $P$ can be simplified to $(1 + x + x^2 + x^3 + x^4 + x^5)^2$? Is it a multinomial thing (which seems a bit too complicated for someone to "notice"), or is there something simpler about the polynomial that one could use to factor it?
 [/INST] Use the distributive property.
$$(1 + x + x^2 + x^3 + x^4 + x^5)(1 + x + x^2 + x^3 + x^4 + x^5) $$
is equal to
$$
    \begin{matrix}
    (1)(1 + x + x^2 + x^3 + x^4 + x^5) \\
     (x)(1 + x + x^2 + x^3 + x^4 + x^5) \\
     (x^2)(1 + x + x^2 + x^3 + x^4 + x^5) \\
     (x^3)(1 + x + x^2 + x^3 + x^4 + x^5) \\
     (x^4)(1 + x + x^2 + x^3 + x^4 + x^5) \\
     (x^5)(1 + x + x^2 + x^3 + x^4 + x^5) \\ 
    \end{matrix}
$$
When the above is expanded, we get $36$ products. Arrange the results of these $36$ products into columns whose entries have the same degree.
$$
    \begin{matrix}
    1 & x & x^2 & x^3 & x^4 & x^5 \\
     & x & x^2 & x^3 & x^4 & x^5 & x^6 \\
     &  & x^2 & x^3 & x^4 & x^5 & x^6 & x^7 \\
     &  & & x^3 & x^4 & x^5 & x^6 & x^7 & x^8 \\
     &  & & & x^4 & x^5 & x^6 & x^7 & x^8 & x^9 \\
     &  & & & & x^5 & x^6 & x^7 & x^8 & x^9 & x^{10} \\ 
    \end{matrix}
$$
Adding like powers of $x$ shows how the pattern of coefficients arises, giving
$$= \;\;\; 1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 5x^6 + 4x^7 + 3x^8 + 2x^9 + x^{10} $$
In the case of $\;(1 + x + x^2 + \ldots + x^{n-1} + x^n)^2,\;$ the same pattern of coefficients --- increasing by increments of $1$ from $1$ to some maximum value, and then decreasing by increments of $1$ from the maximum value to $1$ --- can be seen by thinking about the following.
$$
    \begin{matrix}
    1 & x & x^2 & x^3 & x^4 & \ldots & x^{n-1} & x^n \\
     & x & x^2 & x^3 & x^4 & \ldots & x^{n-1} & x^n & x^{n+1} \\
     &  & x^2 & x^3 & x^4 & \ldots & x^{n-1} & x^n & x^{n+1} & x^{n+2} \\
     &  & & x^3 & x^4 & \ldots & x^{n-1} & x^n & x^{n+1} & x^{n+2} & x^{n+3} \\
     &  & & & x^4 & \ldots & x^{n-1} & x^n & x^{n+1} & x^{n+2} & x^{n+3} & x^{n+4} \\
    \end{matrix}
$$
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</s>