1. **State the problem:** Simplify or express the sum $a^4 + b^4$ in a useful form. 2. **Formula and rules:** There is no simple factorization over real numbers for $a^4 + b^4$ like the difference of squares, but it can be factored over complex numbers or expressed using sums of squares. 3. **Intermediate work:** One useful factorization is: $$a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2$$ which can be rewritten as: $$a^4 + b^4 = (a^2 + b^2 - \sqrt{2}ab)(a^2 + b^2 + \sqrt{2}ab)$$ 4. **Explanation:** This shows $a^4 + b^4$ as a difference of squares of $a^2 + b^2$ and $\sqrt{2}ab$. This factorization is useful in some algebraic contexts but does not factor into linear terms with real coefficients. **Final answer:** $$a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2 = (a^2 + b^2 - \sqrt{2}ab)(a^2 + b^2 + \sqrt{2}ab)$$