.
Let w be the width.
Then the length is (w+5) and the height is (w-4).
The equation for the volume is
w*(w+5)*(w-4) = 416.
Notice that 416 = 8*13*4 - I decomposed the number 416 into the product of 3 factors.
It can be done by different ways, but I took the care that the second factor is 5 more than the first and the third is 4 less than the first -
exactly as the numbers w, (w+5) and (w-4) in the left side of equation (1).
It gives me a confidence that I correctly guessed the solution.
The plot below shows that the guessed solution w = 8 is UNIQUE.
Plot y = x*(x+5)*(x-4) (red) and y = 416 (green)
ANSWER. The width is 8 inches; the length is 8+5 = 13 inches; the height is 8-4 = 4 inches.
Solved, explained and completed.
Probably, there are other ways to find the solution algebraically on more strict base, but what is done is enough to me . . .