Let x represent the width, and y represent the height. Then the volume equation is
.. x*3x*y = 120
.. x^2*y = 40
And the surface area equation is
.. 2(x*3x +y(x +3x)) = 184
.. 3x^2 +4xy = 92
Using the first equation (y = 40/x^2) in the second, we have
.. 3x^2 +4x(40/x^2) = 92
.. 3x^3 -92x +160 = 0 . . . . . after subtracting 92 and multiplying by x.
A graphing calculator shows this has a root at x=2. Factoring that out gives
.. (x -2)(3x^2 +6x -80) = 0
The quadratic has roots
.. x = -1 ±√(83/3) ≈ -6.25991 and 4.25991
Box dimensions may be
.. width: 2 inches
.. length: 6 inches
.. height: 10 inches
or
.. width: 4.25991 inches
.. length: 12.77973 inches
.. height: 2.20424 inches