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

To simplify the expression \( (2p + q)^3 - (p - q)^3 \), we can use the difference of cubes formula, which states that \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Let \( a = 2p + q \) and \( b = p - q \). First, we find \( a - b \): \[ a - b = (2p + q) - (p - q) = 2p + q - p + q = p + 2q \] Next, we need to calculate \( a^2 + ab + b^2 \): 1. \( a^2 = (2p + q)^2 = 4p^2 + 4pq + q^2 \) 2. \( b^2 = (p - q)^2 = p^2 - 2pq + q^2 \) 3. \( ab = (2p + q)(p - q) = 2p^2 - 2pq + pq - q^2 = 2p^2 - pq - q^2 \) Now summing these: \[ a^2 + ab + b^2 = (4p^2 + 4pq + q^2) + (2p^2 - pq - q^2) + (p^2 - 2pq + q^2) \] Combining like terms: \[ (4p^2 + 2p^2 + p^2) + (4pq - pq - 2pq) + (q^2 - q^2 + q^2) = 7p^2 + pq + q^2 \] Finally, substituting back into our formula gives: \[ (2p + q)^3 - (p - q)^3 = (p + 2q)(7p^2 + pq + q^2) \] So the simplified form of the expression \( (2p + q)^3 - (p - q)^3 \) is: \[ (p + 2q)(7p^2 + pq + q^2) \]