Factor \( 27 h^{3}-125 d^{3} \) completely.

To factor the expression \( 27 h^{3}-125 d^{3} \), we can recognize this as a difference of cubes. The formula for factoring a difference of cubes is: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] In this case, we have: \[ 27 h^{3} = (3h)^3 \quad \text{and} \quad 125 d^{3} = (5d)^3 \] So, we can set \( a = 3h \) and \( b = 5d \). Now we can apply the difference of cubes formula: 1. The first factor is \( a - b = 3h - 5d \). 2. The second factor is \( a^2 + ab + b^2 = (3h)^2 + (3h)(5d) + (5d)^2 = 9h^2 + 15hd + 25d^2 \). Putting it all together gives us: \[ 27 h^{3} - 125 d^{3} = (3h - 5d)(9h^2 + 15hd + 25d^2) \] Thus, the complete factored form of the expression is: \[ \boxed{(3h - 5d)(9h^2 + 15hd + 25d^2)} \]