1. The problem is to verify if the equation $$p^4 - q^4 = (p - q)(p^3 + p^2q + pq^2 + q^3)$$ is a polynomial identity. 2. Recall the difference of fourth powers factorization: $$a^4 - b^4 = (a - b)(a^3 + a^2b + ab^2 + b^3)$$ which is a known identity. 3. Substitute $a = p$ and $b = q$ into the formula: $$p^4 - q^4 = (p - q)(p^3 + p^2q + pq^2 + q^3)$$ 4. To verify, expand the right side: $$\begin{aligned} (p - q)(p^3 + p^2q + pq^2 + q^3) &= p \cdot (p^3 + p^2q + pq^2 + q^3) - q \cdot (p^3 + p^2q + pq^2 + q^3) \\ &= p^4 + p^3q + p^2q^2 + pq^3 - (qp^3 + qp^2q + qpq^2 + q^4) \\ &= p^4 + p^3q + p^2q^2 + pq^3 - p^3q - p^2q^2 - pq^3 - q^4 \\ &= p^4 - q^4 \end{aligned}$$ 5. Notice the terms $p^3q$, $p^2q^2$, and $pq^3$ cancel out with their negatives. 6. Therefore, the equation holds true for all values of $p$ and $q$, confirming it is a polynomial identity. Final answer: $$p^4 - q^4 = (p - q)(p^3 + p^2q + pq^2 + q^3)$$ is a polynomial identity.