One of the factors of $\left|\begin{array}{cccc}a& a& a& a\\ b& b& b& b\\ c& c& c& c\\ d& d& d& d\end{array}\right|$  is

$\left|\begin{array}{cccc}a& a& a& a\\ b& b& b& b\\ c& c& c& c\\ d& d& d& d\end{array}\right|$

Operating $C_1\to C_1+C_2+C_3+C_4$ , we find that $a+b+c+d$  can be taken out from $C_1$ 

$\Rightarrow a+b+c+d$  is a factor of the given determinant.