Find the LCM of the expressions: (x^2 - y^2 - z^2 - 2yz), (x^2 - y^2 + z^2 + 2xz), and (x^2 + y^2 - z^2 - 2xy). Express the LCM as a product of linear factors.

Introduction:
The task is to find the least common multiple of three quadratic forms by factoring each into linear factors and then collecting distinct factors with the highest needed multiplicities.

Given Data / Assumptions:

• f1 = x^2 - y^2 - z^2 - 2yz
• f2 = x^2 - y^2 + z^2 + 2xz
• f3 = x^2 + y^2 - z^2 - 2xy

Concept / Approach:
Rewrite each quadratic as a difference of squares by grouping. Factor each to identify the linear factors present across the set. The LCM collects each distinct factor at the maximum multiplicity it occurs in any single polynomial.

Step-by-Step Solution:

Verification / Alternative check:
Each of f1, f2, and f3 divides the product (x + y + z)(x - y - z)(x - y + z). No other linear factor is required.

Why Other Options Are Wrong:
Options that replace (x - y - z) with (x + y - z) or omit factors do not contain all necessary linear factors to cover each polynomial.

Common Pitfalls:
Sign errors when grouping terms to form a difference of squares, which can flip the identification of factors.

Final Answer:
(x + y + z)(x - y - z)(x - y + z)