Factorizing (2a-b-c)^3 (2b-c-a)^3 (2c-a-b)^3

To factorize the given expression, we can make use of the algebraic identity for the cube of a binomial:

(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

Applying this identity to each term in the expression, we can expand it and then combine like terms to find the factorized form.

Expanding the expression:

(2a - b - c)^3 = (2a)^3 - 3(2a)^2(b) + 3(2a)(b^2) - (b - c)^3
= 8a^3 - 12a^2b + 6ab^2 - b^3 - 6a^2c + 6abc - 3b^2c + 3ac^2 - 3bc^2 + c^3

(2b - c - a)^3 = (2b)^3 - 3(2b)^2(c) + 3(2b)(c^2) - (c - a)^3
= 8b^3 - 12b^2c + 6bc^2 - c^3 - 6ab^2 + 6abc - 3ac^2 + 3b^2a - 3ba^2 + a^3

(2c - a - b)^3 = (2c)^3 - 3(2c)^2(a) + 3(2c)(a^2) - (a - b)^3
= 8c^3 - 12c^2a + 6ca^2 - a^3 - 6bc^2 + 6abc - 3ab^2 + 3c^2b - 3cb^2 + b^3

Combining like terms:

Now, we can combine like terms to simplify the expression:

8a^3 - 12a^2b + 6ab^2 - b^3 - 6a^2c + 6abc - 3b^2c + 3ac^2 - 3bc^2 + c^3
+ 8b^3 - 12b^2c + 6bc^2 - c^3 - 6ab^2 + 6abc - 3ac^2 + 3b^2a - 3ba^2 + a^3
+ 8c^3 - 12c^2a + 6ca^2 - a^3 - 6bc^2 + 6abc - 3ab^2 + 3c^2b - 3cb^2 + b^3

Simplifying the expression:

By rearranging the terms, we can combine like terms:

(8a^3 + a^3 - a^3) + (-12a^2b + 6ab^2 + 6abc - 6ab^2 - 3ba^2)
+ (6ab^2 - 6ab^2 + 6abc -