# A rectangular prism has a volume of V= 2x^3+6x^2+4x. What could the dimensions of this prism be?

Jul 29, 2015

$2 {x}^{3} + 6 {x}^{2} + 4 x = 2 x \left(x + 1\right) \left(x + 2\right)$

So some possible dimensions are:

$2 x \times \left(x + 1\right) \times \left(x + 2\right)$

$x \times 2 \left(x + 1\right) \times \left(x + 2\right)$

$x \times \left(x + 1\right) \times 2 \left(x + 2\right)$

#### Explanation:

$2 {x}^{3} + 6 {x}^{2} + 4 x = 2 x \left({x}^{2} + 3 x + 2\right) = 2 x \left(x + 1\right) \left(x + 2\right)$

I find these kind of questions rather strange, as they place no real limitations on the dimensions apart from the total volume.

In fact, given any $a , b > 0$, let $c = \frac{2 {x}^{3} + 6 {x}^{2} + 4 x}{a b}$

Then a rectangular prism of sides $a$, $b$ and $c$ has the required volume.