1. The problem is to understand the rule for expanding $a^4 + b^4$. 2. The expression $a^4 + b^4$ is a sum of fourth powers, which does not factor using the simple rule of adding one to $b$ and subtracting from $a$. 3. Important rule: Unlike $a^2 - b^2 = (a-b)(a+b)$, the sum of even powers like $a^4 + b^4$ does not factor over the real numbers into linear factors. 4. However, $a^4 + b^4$ can be factored over complex numbers or expressed as: $$a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2 = (a^2 + b^2 - \sqrt{2}ab)(a^2 + b^2 + \sqrt{2}ab)$$ 5. This is a difference of squares factorization applied cleverly. 6. So the rule you mentioned (adding one to $b$ and subtracting from $a$) is not correct for $a^4 + b^4$. 7. Summary: $a^4 + b^4$ does not factor simply by adjusting exponents; it requires special factorization techniques or remains unfactored in basic algebra.