To complete the factorization of the expression
\[ (a^2 + b^2 - c^2)^2 - 4a^2 b^2 - 4a^2 c^2 + 4b^2 c^2, \]
we can simplify it step by step.
Let's first recognize that the expression can be potentially factored by rearranging or regrouping it. It takes a form that can be examined for known identities or factorable patterns.
We can rewrite the expression:
\[ (a^2 + b^2 - c^2)^2 - (2ab)^2 - (2ac)^2 + (2bc)^2. \]
Recognizing this resembles a squared term minus a sum of squared terms, we can see if it matches any known identities.
After simplification and manipulation, one can check that the expression yields:
\[ (a + b - c)^2 (a - b + c)^2. \]
Thus we find that it can indeed be expressed in terms of products of simpler binomials.
Now, to write down the correct factorization in the specified form:
The expression can be factored as:
\[ (a + b - c)(a + b - c)(a - b + c)(a - b + c). \]
So, filling in the blanks in the requested format gives us:
\[ (a + b - c)(a + b - c)(a - b + c)(a - b + c). \]
Therefore, the filled-in blanks are:
\( b - c \) in the first and second blanks, and \( -b + c \) in the third and fourth blanks.
Thus, the full factorization expression to fill the blanks is:
\[ (a + b - c)(a + b - c)(a - b + c)(a - b + c). \]
Final answer:
\[ (b - c), (b - c), (-b + c), (-b + c). \]
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