How did people come up with the formula $a^3+b^3=(a+b)(a^2-ab+b^2)$? Every resource that I've read proves the formula
$$ a^3 + b^3 = (a+b)(a^2-ab+b^2) \tag1$$
by just multiplying $(a+b)$ and $(a^2 - ab + b^2)$.
But how did people come up with that formula? Did they think like, "Oh, let's just multiply these polynomials, I don't know why, let's just do it." I don't think that people just pointed a finger at the sky and came up with that formula.
So, how to prove $(1)$?

$«$Who first factored the expression $a^3+b^3$ and what was the method used?$»$ - I don't know the exact answer to this question, but the similar question I'm trying to answer is:
$«$How can we factor $a^3+b^3$ using the most basic algebraic techniques?$»$
It seems to me that, this formula basically comes from the Binomial theorem:
$$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$
But, without the binomial theorem you can obtain this result as follows:
$$\begin{aligned}(a+b)^3&=(a+b) \left((a+b)\times (a+b)\right)\\
&=(a+b)(a^2+2ab+b^2)\\
&=a^3+3a^2b+3ab^2+b^3\end{aligned}$$
Then observe that,
$$\begin{aligned}a^3+3a^2b+3ab^2+b^3&=(a^3+b^3)-3ab(a+b)\end{aligned}$$
This leads,
$$
\begin{aligned}a^3+b^3&=(a+b)^3-3ab(a+b)\\
&=(a+b)\left((a+b)^2-3ab\right)\\
&=(a+b)(a^2-ab+b^2).\end{aligned}
$$
