How do we solve this word problem by using polynomial function?

Question

Cindy works at Jurassic Park and has been tasked to design a container in the shape of a rectangular prism for the incoming baby dinosaurs. The scaled model of the container has dimensions 2m by 4m by 6m. Cindy has decided to increase each dimension of the scaled model by the same amount in order to produce a container with a volume of 84 times the volume of the scale model. By what amount should Cindy increase each dimension of the scaled model? [5 marks]

Answer 1

2x4x6 = 48

(2+x)(4+x)(6+x) = 84

(x^2 + 6x +8)(x+6) = 84

x^3 +12x^2 + 44x + 48 = 84

x^3 + 12x^2 + 44x - 36= 0

only 1 possible positive solution for x using Descartes Rule of Signs

2 possible negative real solutions, but probably both imaginary

which means one real positive solution

x= 0 V =2(4)(6) = 48

x=1 V=3(5)(7) = 105

x has to be between 0 and 1, some fraction of a meter

x=1/2, V = (5/2)(9/2)(13/2) = 73 1/8

x has to be between 1/2 and 1

1/2<x<1

x=3/4, V = (11/4)(19/4)(27/4) = 88 11/64

1/2 < x < 3/4

cubic equations are difficult to solve

but

using an online equation solver

x = cube root of (2sqr11891/3^(3/2) +42) + 4/3cube root of (2sqr11891/3^(3/2)-42) - 4