Final answer:
The expression representing the height of the box is (x^3 - 63x + 162) / (x - 3)(x + 9).
Step-by-step explanation:
The volume of a rectangular prism is given by the formula V = length x width x height. In this case, the volume is represented by the function x^3 - 63x + 162. The length of the box is x - 3 and the width of the box is x + 9.
To find the expression representing the height, we can divide the volume function by the product of the length and width. So, the expression representing the height of the box is:
Height = (x^3 - 63x + 162) / (x - 3)(x + 9)