1. **State the problem:** Simplify the expression \( \frac{A}{B} \) where \[ A = \frac{(p - q)^2 + 4pq}{p^3 - q^3 - 3pq(p - q)} \quad \text{and} \quad B = \frac{p^3 + q^3 + 3pq(p + q)}{(p + q)^2 - 4pq} \] 2. **Rewrite the division as multiplication by the reciprocal:** \[ \frac{A}{B} = A \times \frac{1}{B} = \frac{(p - q)^2 + 4pq}{p^3 - q^3 - 3pq(p - q)} \times \frac{(p + q)^2 - 4pq}{p^3 + q^3 + 3pq(p + q)} \] 3. **Simplify the numerators and denominators separately:** - Expand \((p - q)^2 + 4pq\): \[ (p - q)^2 + 4pq = (p^2 - 2pq + q^2) + 4pq = p^2 + 2pq + q^2 = (p + q)^2 \] - Expand \(p^3 - q^3 - 3pq(p - q)\): Recall \(p^3 - q^3 = (p - q)(p^2 + pq + q^2)\), so \[ p^3 - q^3 - 3pq(p - q) = (p - q)(p^2 + pq + q^2) - 3pq(p - q) = (p - q)(p^2 + pq + q^2 - 3pq) \] Simplify inside the parentheses: \[ p^2 + pq + q^2 - 3pq = p^2 - 2pq + q^2 = (p - q)^2 \] So denominator becomes: \[ (p - q)(p - q)^2 = (p - q)^3 \] - Expand \((p + q)^2 - 4pq\): \[ (p + q)^2 - 4pq = (p^2 + 2pq + q^2) - 4pq = p^2 - 2pq + q^2 = (p - q)^2 \] - Expand \(p^3 + q^3 + 3pq(p + q)\): Recall \(p^3 + q^3 = (p + q)(p^2 - pq + q^2)\), so \[ p^3 + q^3 + 3pq(p + q) = (p + q)(p^2 - pq + q^2) + 3pq(p + q) = (p + q)(p^2 - pq + q^2 + 3pq) \] Simplify inside the parentheses: \[ p^2 - pq + q^2 + 3pq = p^2 + 2pq + q^2 = (p + q)^2 \] So numerator becomes: \[ (p + q)(p + q)^2 = (p + q)^3 \] 4. **Substitute the simplified expressions back:** \[ \frac{A}{B} = \frac{(p + q)^2}{(p - q)^3} \times \frac{(p - q)^2}{(p + q)^3} = \frac{(p + q)^2 (p - q)^2}{(p - q)^3 (p + q)^3} \] 5. **Simplify the fraction:** \[ = \frac{(p + q)^2}{(p + q)^3} \times \frac{(p - q)^2}{(p - q)^3} = \frac{1}{p + q} \times \frac{1}{p - q} = \frac{1}{(p + q)(p - q)} \] 6. **Final answer:** \[ \boxed{\frac{1}{(p + q)(p - q)}} \]