<s>[INST] How can one find the factorization $a^4 + 2a^3 + 3a^2 + 2a + 1 = (a^2 + a + 1)^2$ from scratch? I want to prove $a^4 + 2a^3 + 3a^2 + 2a + 1$ always be perfect square for $a \in \mathbb{N}$. Using GeoGebra I found $a^4 + 2a^3 + 3a ² + 2a + 1 = (a^2 + a + 1)^2$ which is trivial to verify.
Since this equation has no real roots, it's not so easy to decompose it through division by $a-a_0$ where $a_0$ is a root. I don't want to use complex numbers if possible. How do find this factorization from scratch / without using a computer?
 [/INST] The coefficients $1,2,3,2,1$ are suggestive of the following:
$$\begin{array}{c}  & 1a^4 & + & 2a^3 & + & 3a^2 & + & 2a & + & 1 \\ \hline
= & a^4 & + & a^3 & + & a^2 \\ & & + & a^3 & + & a^2 & + & a \\ & & & & + & a^2 & + & a & + & 1 \end{array} $$
$$\begin{array}{l} = & a^2(a^2+a+1)+a(a^2+a+1)+1(a^2+a+1) \\ = & (a^2+a+1)^2. \end{array}$$
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