1. The problem is to factor the expression $a^4 - b^4$. 2. This is a difference of squares because $a^4 = (a^2)^2$ and $b^4 = (b^2)^2$. 3. The difference of squares formula is $x^2 - y^2 = (x - y)(x + y)$. 4. Applying this to $a^4 - b^4$, we get: $$a^4 - b^4 = (a^2)^2 - (b^2)^2 = (a^2 - b^2)(a^2 + b^2)$$ 5. Notice $a^2 - b^2$ is also a difference of squares, so factor it further: $$a^2 - b^2 = (a - b)(a + b)$$ 6. Therefore, the full factorization is: $$a^4 - b^4 = (a - b)(a + b)(a^2 + b^2)$$ 7. This is the simplest factorization over the real numbers.