Expressing a $3\times 3$ determinant as the product of four factors I am attempting to express the determinant below as a product of four linear factors
$$\begin{vmatrix}
a & bc & b+c\\
b & ca & c+a\\
c & ab & a+b\\
\end{vmatrix}
= 
a\begin{vmatrix}
ca & c+a\\
ab & a+b\\
\end{vmatrix}
-
bc\begin{vmatrix}
b & c+a\\
c & a+b\\
\end{vmatrix}
+(b+c)\begin{vmatrix}
b & ca\\
c & ab\\
\end{vmatrix}$$
This is as far as I get before it gets too messy
$$
=a^3(c-b)-bc\{(b-c)(b+c)+a(b-c)\}+a(b-c)(b+c)^2
$$
But I cant seem to arrive at the answer in the book, which is given as 
$$
(a-b)(b-c)(c-a)(a+b+c)
$$
Am I doing something wrong as I have been stuck on this question for three days.
Thanks in advance!
$$
$$

Start by adding the 1st to the 3rd column to create a column of $a+b+c$'s. Then subtract 3rd row from 2nd & 1st ones to make two out of three entries in that column zero. Now expand wrt that column.
