Answer: Let's call the length of the box "x". Then, the height of the box is 4x and the width is x + 7. The volume of the box is given by the equation:
x * 4x * (x + 7) = 102
Expanding and simplifying the equation, we get:
4x^3 + 28x^2 - 102 = 0
To find the length of the box, we need to solve for x in this equation. One way to do this is to use the cubic formula. However, this formula can be quite complex, so a simpler method would be to use numerical methods such as the bisection method or Newton's method.
Explanation: