This post is inspired by Prove that the polynomial $\prod\limits_{i=1}^n\,\left(x-a_i\right)-1$ is irreducible in $\mathbb{Z}[x]$..
(A) Find all positive integers $n$ and integers $a_1,a_2,\ldots,a_n$ such that $a_1< a_2 < \ldots < a_n$ and that the polynomial $$f(x):=\left(x-a_1\right)\left(x-a_2\right)\cdots\left(x-a_n\right)+1$$ is a reducible element of $\mathbb{Z}[x]$ (or equivalently, of $\mathbb{Q}[x]$).
To prove (A), one attempt is to show that $f(x)$ can not be a perfect square. Hence, it is natural to ask the following related problem (B).
(B) Find all positive integers $n$ and integers $a_1,a_2,\ldots,a_n$ such that $a_1 \leq a_2 \leq \ldots \leq a_n$ and that the polynomial $$f(x):=\left(x-a_1\right)\left(x-a_2\right)\cdots\left(x-a_n\right)+1$$ is a perfect square in $\mathbb{Z}[x]$ (or equivalently, in $\mathbb{Q}[x]$).
While I have solutions to (A) and (B), I don't yet have a solution to the problem (C) given below.
(C) Find all positive integers $n$ and integers $a_1,a_2,\ldots,a_n$ such that $a_1 \leq a_2 \leq \ldots \leq a_n$ and that the polynomial $$f(x):=\left(x-a_1\right)\left(x-a_2\right)\cdots\left(x-a_n\right)+1$$ is reducible in $\mathbb{Z}[x]$ (or equivalently, in $\mathbb{Q}[x]$).
I also don't have a solution to this modified version (D) of the link above.
(D) Find all positive integers $n$ and integers $a_1,a_2,\ldots,a_n$ such that $a_1 \leq a_2 \leq \ldots \leq a_n$ and that the polynomial $$f(x):=\left(x-a_1\right)\left(x-a_2\right)\cdots\left(x-a_n\right)-1$$ is reducible in $\mathbb{Z}[x]$ (or equivalently, in $\mathbb{Q}[x]$).
Even more generally, we have this seemingly monstrously difficult problem (E).
(E) For given $u\in\mathbb{N}$ and $v\in\mathbb{Z}$ with $\gcd(u,v)=1$, find all integers $n$ and integers $a_1,a_2,\ldots,a_n$ such that $a_1 \leq a_2 \leq \ldots \leq a_n$ and that the polynomial $$f(x):=u\left(x-a_1\right)\left(x-a_2\right)\cdots\left(x-a_n\right)+v$$ is reducible in $\mathbb{Z}[x]$ (or equivalently, in $\mathbb{Q}[x]$).
If (E) seems too difficult, it may make sense to try when $u$ and $|v|$ are perfect $k$-th powers for some integer $k>1$, when $u$ and $v$ are prime integers or $\pm 1$, or when $a_1<a_2<\ldots<a_n$ is assumed.
I know that this post is loaded with multiple questions. However, as they are very closely related, I don't think it hurts to place them in the same thread.
P.S. Part (A) already has an answer here: Proving that a polynomial of the form $(x-a_1)\cdots(x-a_n) + 1$ is irreducible over $\mathbb{Q}$.
