Expanding the Expression (a+b+c)(b+ca)(c+ab)(a+bc)
This expression looks daunting at first glance, but with a clever trick, we can simplify it significantly. Let's break down the solution stepbystep.
Understanding the Pattern
Notice that each factor in the expression has a unique pattern:
 First factor: (a + b + c)
 Second factor: (b + c  a)
 Third factor: (c + a  b)
 Fourth factor: (a + b  c)
Observe that each factor has two positive terms and one negative term, and the sign of each variable changes consistently. This pattern is key to our solution.
The Trick
Let's introduce a new variable, 'x', and rewrite the expression using this variable:
1. Substitute:
 Let x = a + b + c
Now, we can express the factors in terms of 'x':
 (a + b + c) = x
 (b + c  a) = x  2a
 (c + a  b) = x  2b
 (a + b  c) = x  2c
2. Substitute in the original expression:
Our original expression now becomes:
(a + b + c)(b + c  a)(c + a  b)(a + b  c) = x(x  2a)(x  2b)(x  2c)
Expanding the Simplified Expression
We can now easily expand the simplified expression:

Multiply the first two factors:
x(x  2a) = x²  2ax

Multiply the last two factors:
(x  2b)(x  2c) = x²  2bx  2cx + 4bc

Multiply the results from step 1 and step 2:
(x²  2ax)(x²  2bx  2cx + 4bc) = x⁴  2ax³  2bx³ + 4bcx²  2cx³ + 4acx² + 4abx²  8abcx

Simplify by combining like terms:
x⁴  2(a + b + c)x³ + 4(ab + ac + bc)x²  8abcx

Substitute back x = a + b + c:
(a + b + c)⁴  2(a + b + c)³ + 4(ab + ac + bc)(a + b + c)²  8abc(a + b + c)
The Final Solution
Therefore, the expanded form of (a + b + c)(b + c  a)(c + a  b)(a + b  c) is:
(a + b + c)⁴  2(a + b + c)³ + 4(ab + ac + bc)(a + b + c)²  8abc(a + b + c)
This solution, while seemingly complex, is much easier to manage than the original expression. The key was introducing a new variable 'x' to simplify the pattern and make the expansion process more manageable.