Factoring $1+x+\dots +x^n$ into a product of polynomials with positive coefficients

Can the polynomial $1+x+x^2+\dots +x^n$ be factored, for some $n\ge 1$, into a product of two non-constant polynomials with positive coefficients?

$$x^n+\cdots +x+1=\frac{x^{n+1}-1}{x-1}=\frac{\prod_{1\le k\le n+1}(x - e^{2i\pi k/(n+1)})}{x-1}=\prod_{1\le k\lt n+1}(x - e^{2i\pi k/(n+1)})$$ For a factorization $x^n+\dots +x+1=g(x)h(x)$, $g$ and $h$ are products of some $x-\zeta^k$ where $\zeta$ is nth root of unity (up to units (in this case multiplying by positive real numbers) however any such factorization implies one without units). The leading term of both polynomials must be $1$ because it is the product of leading terms that are always $1$. The constant term is also one because it is the product of root of unity so its absolute value is $1$ and it must be real and positive.

$$(x^p+\cdots +1)(x^q+\cdots+1)=x^n+\cdots +x+1$$ Assume without loss of generality that $p\ge q$. Take the constant term of the first polynomial and multiply it with the leading term of the second and you get $x^q$. $$\begin{align} (x^p+\cdots+ ax^q +\cdots +1)(x^q+\cdots+1) &=\cdots +ax^q1+ \cdots +1x^q +\cdots\\ &= \cdots +(\cdots+a+1)x^q \\ \end{align}$$ All other terms must be positive and they can only add to $x^q$, the coefficient should be equal $1$. Therefore all other coefficients of $x^q$ must be zero and $a$ is zero.