let

x = the width of the box, x>0

x + 1 = the height of the box

x + 9 = the length of the box

V = 420 in^3 the volume of the box

V = width*length*height

V = x(x + 1)(x + 9)

x(x + 1)(x + 9) = 420

x(x + 1)(x + 9) - 420 = 0

by factoring the last equation we get:

........

click here to see the step by step factorization of the polynomial

........

(x - 5)(x^2 + 15x + 84) = 0

One root is x = 5.

By solving x^2 + 15x + 84 = 0 we find the other two roots:

........

click here to see all the equation solution steps

........

The roots are complex numbers which leaves us with the only possible solution x = 5 in.

x + 1 = 5 + 1 = 6 in

x + 9 = 5 + 9 = 14 in

The dimensions of the box are: the width is 5 inches, the length is 14 inches and the height is 6 inches.