How should I calculate the determinant? $\left|\begin{array}{cccc}1&a&b&c+d\\1&b&c&a+d\\1&c&d&a+b\\1&d&a&b+c\end{array}\right|=
\left|\begin{array}{cccc}1&a&b&c\\1&b&c&a\\1&c&d&a\\1&d&a&b\end{array}\right|+
\left|\begin{array}{cccc}1&a&b&d\\1&b&c&d\\1&c&d&b\\1&d&a&c\end{array}\right|$
I tried to calculate the determinant but I couldn't do it after separating the determinant by the property.  How should I calculate it?  

$${\begin{vmatrix}1&a & b &c+d\\1 &b &c &d+a \\1 &c &d &a+b\\1&d &a &b+c &\end{vmatrix}} \space  c_2+c_3+c_4 \to c_4 \\
{\begin{vmatrix}1&a & b &a+b+c+d\\1 &b &c &a+b+c+d \\1 &c &d &a+b+c+d\\1&d &a &a+b+c+d &\end{vmatrix}} \space factor  \space (a+b+c+d)=\\(a+b+c+d)
{\begin{vmatrix}1&a & b &1\\1 &b &c &1 \\1 &c &d &1\\1&d &a &1 &\end{vmatrix}}$$now $c_1-c_4 \to c_1$
$${\begin{vmatrix}0&a & b &1\\0 &b &c &1 \\0 &c &d &1\\0&d &a &1 &\end{vmatrix}} $$
