How to solve the following quadratic word problem? The total cost of carpeting a rectangular room is given the expression $$6x^2 + 18x$$
This is the multiple choice type question so the given options were set up like this. 
The length of the room is_______feet, its width is ____ feet and the cost of carpeting is _____ per square. 
(i'm purposely leaving them as blank because I want to know how to solve it. 
My question is how would i find the blank parts?
if i factor the equation, i'll get 
$$ 6x(x + 3)$$
this wouldn't give me any information about the blanks above?
What would I do?    

Steve X is correct that there are infinitely many solutions.  
But let's consider what is the "best" solution.  It is perfectly natural in a word problem that the unknown $x$ refers to one of the properties of the physical situation (i.e., length or width or price per square foot).  Once we factor the polynomial (but see below):
$6 x (x+3)$ 
we can assume that each of the three terms refers to one of the properties of the physical situation.  If $x$ referred to the cost per square foot, it would make no physical sense that the width or length would be $x + 3$.  That is, if $x$ is in dollars per square foot, then $x + 3$ would not have the units of a length.
Both length and width have the same units (e.g., feet or meters), so it is most natural that $x$ refers to width and thus $x + 3$ refers to length (which is typically longer than width).
So I think the best solution is:  


*

*Width of room = $x$

*Length of room = $x+3$

*Cost per unit area = 6


Note that there are an infinite number of ways to factor the given equation into three terms, even if we assume that $x$ is a "natural" factor:
$x (x+3) 6$
$x (2 x + 6) 3$
$x (a x + 3 a) (6/a)$ for any real positive $a$.
There are limits on $a$.  For example it would make no sense for any property in this problem to have a negative value, and thus $a > 0$.  Likewise, it makes no sense for the area of the rug to be extraordinarily large (e.g., larger than the Atlantic Ocean), so there are limits on $a$.
Regardless, as Steve X points out:  there are an infinite number of solutions.
