What are the possible expressions for the dimensions of the cuboids whose volumes are given as below?
I. $ 3{x^2} - 12x $
II. $ 12k{y^2} + 8ky - 20k $

Hint: Since, the volume of a cuboid is the product of length, breadth and height therefore by factorizing the given volume, we can know the length, breadth and height of the cuboid.

Complete step-by-step answer:
As we know that volume of a cuboid is the product of length, breadth and height so from the given expressions we need to find the individual values of length, breadth and height by factorizing the given expressions.
Expression 1: $ 3{x^2} - 12x $
From the given expression we can understand that x and 3 it a common factor in the both the term i.e $ 3{x^2} $ & $ 12x $ so we need to simply factorise the given expression
 $ 3{x^2} - 12x $
 $ = 3x(x - 4) $
 $ = 3 \times x \times (x - 4) $
Hence, possible expression for length $ = 3 $
$\Rightarrow$ Possible expression for breadth $ = x $
$\Rightarrow$ Possible expression for height $ = (x - 4) $

Expression 2: $ 12k{y^2} + 8ky - 20k $
From the given expression we can understand that 4k is a common factor in the given expression so we need to simply factorise the given expression and then need to split the middle term to find the three values.
 $ 12k{y^2} + 8ky - 20k $
 $ = 4k(3{y^2} + 2y - 5) $
Further we need to factorise the expression $ (3{y^2} + 2y - 5) $ by splitting the middle term.
 $ = 4k(3{y^2} + 2y - 5) $
\[ = 4k(3{y^2} + 5y - 3y - 5)\]
\[ = 4k[y(3y + 5) - 1(3y + 5)]\]
\[ = 4k(3y + 5)(y - 1)\]
Hence, possible expression for length $ = 4k $
$\Rightarrow$ Possible expression for breadth $ = (3y + 5) $
$\Rightarrow$ Possible expression for height $ = (y - 1) $

Note: If in case a negative value gets derived after factorisation, kindly ignore the negative sign. The coordinate will be the value only. And the possible values of length breadth and width can be any one of the three values obtained irrespective of any condition.