I am wondering if a multi-variable polynomial in $k[x_{1},\ldots,x_{n}]$ of two terms $$f(x)=c_{1}\text{monomial}_{1}+c_{2}\text{monomial}_{2}$$ could be factored into a product of two polynomials of more than $2$ terms?
Of course, in $n=1$, there is an example of one of the factors has more than $2$ terms $$x^{p}-1=(x-1)(x^{p-1}+\cdots+1).$$ But I don't think in $n=1$ one can get both factors having more than $2$ terms (even after a possible field extension). So I suspect in large $n$ case, it is also not possible?
An example of such factorization for $\,n=2\,$ is the Sophie Germain identity:
$$ x_1^4 + 4 x_2^4 = \left(x_1^2+2x_1x_2+2x_2^2\right) \left(x_1^2-2x_1x_2+2x_2^2\right) $$
For the general case, $\,x_1,x_2\,$ can be replaced with arbitrary monomials in the $\,n\,$ variables.
[ EDIT ] Other examples can be derived from cyclotomic polynomial identities e.g.
$$ x^9 - 1 = \left(x-1\right)\left(x^2+x+1\right) \left(x^6+x^3+1\right) $$