18. \( (2 p+q)^{3}-(p-q)^{3} \)

To simplify the expression \( (2p+q)^{3}-(p-q)^{3} \), we can apply the identity for the difference of cubes, which states that \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \). Here, let \( a = 2p + q \) and \( b = p - q \). Therefore, we can compute: 1. **Calculate \( a - b \):** \[ a - b = (2p + q) - (p - q) = 2p + q - p + q = p + 2q. \] 2. **Calculate \( a^2 + ab + b^2 \):** - First, find \( a^2 = (2p + q)^2 = 4p^2 + 4pq + q^2 \). - Next, find \( b^2 = (p - q)^2 = p^2 - 2pq + q^2 \). - Now, find \( ab = (2p + q)(p - q) = 2p^2 - 2pq + pq - q^2 = 2p^2 - pq - q^2 \). - Combine \( a^2 + ab + b^2 \): \[ a^2 + ab + b^2 = (4p^2 + 4pq + q^2) + (2p^2 - pq - q^2) + (p^2 - 2pq + q^2) = 7p^2 + pq + 4pq = 7p^2 + 3pq. \] Putting it all together, we get: \[ (2p+q)^3 - (p-q)^3 = (p + 2q)(7p^2 + 3pq). \] Thus, the final simplified form is: \[ (2p + q)^{3} - (p - q)^{3} = (p + 2q)(7p^2 + 3pq). \]