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)