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 follow these steps:
Step 1: Notice that a^4 + b^4 can be factored as (a^2)^2 + (b^2)^2, which is a sum of squares. This is a special case and can be factored further using the sum of squares formula.
Step 2: Apply the sum of squares formula, which states that a^2 + b^2 = (a + b)(a - b). In this case, let's use a^2 as "a" and b^2 as "b".
Therefore, a^4 + b^4 = (a^2)^2 + (b^2)^2 = (a^2 + b^2)(a^2 - b^2).
Step 3: Simplify the expression.
We now have:
(a^2 + b^2)(a^2 - b^2) + c^2 - 2(a^2 b^2 + a^2 c + b^2 c)
Step 4: Notice that (a^2 - b^2) can be factored using the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b). In this case, let's use a as "a" and b as "b".
Therefore, (a^2 - b^2) = (a + b)(a - b).
Step 5: Write the expression using the factored forms.
We now have:
(a^2 + b^2)(a + b)(a - b) + c^2 - 2(a^2 b^2 + a^2 c + b^2 c)
Step 6: The expression is now factored completely.
The final factored expression is:
(a^2 + b^2)(a + b)(a - b) + c^2 - 2(a^2 b^2 + a^2 c + b^2 c)