Factor:

a^4 + b^4 + c^2 - 2(a^2 b^2 + a^2 c + b^2 c)

To factor the expression a^4 + b^4 + c^2 - 2(a^2 b^2 + a^2 c + b^2 c), we can use the difference of squares and the perfect square trinomial formulas.

First, let's rewrite the original expression as follows:

a^4 + b^4 + c^2 - 2a^2 b^2 - 2a^2 c - 2b^2 c

Now, let's factor the first two terms, a^4 + b^4, as the sum and difference of squares:

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

Next, let's regroup the remaining terms:

[(a^2 + b^2) - 2a^2 b^2] + [c^2 - 2a^2 c - 2b^2 c]

Now, let's factor out common terms from each group:

(a^2 + b^2)(1 - 2a^2 b^2) + c(c - 2a^2 - 2b^2)

Notice that we can factor out -2 from the second term:

(a^2 + b^2)(1 - 2a^2 b^2) + c(-2a^2 - 2b^2 + c)

Finally, let's factor out -2 from the second set of parentheses:

(a^2 + b^2)(1 - 2a^2 b^2) - 2c(a^2 + b^2 - c)

So the factored form of the expression a^4 + b^4 + c^2 - 2(a^2 b^2 + a^2 c + b^2 c) is:

(a^2 + b^2)(1 - 2a^2 b^2) - 2c(a^2 + b^2 - c)