What are the possible expressions for the dimensions of the cuboids whose volumes are given below?
$\text{Volume} : 12ky^{2} + 8ky - 20k$

(A) The volume of a cuboid is given by the formula: $\text{Volume} = \text{Length} \times \text{Breadth} \times \text{Height}$.
Given volume $= 12ky^{2} + 8ky - 20k$.
First,factor out the common term $4k$:
$12ky^{2} + 8ky - 20k = 4k(3y^{2} + 2y - 5)$.
Next,factor the quadratic expression $(3y^{2} + 2y - 5)$ by splitting the middle term:
$3y^{2} + 2y - 5 = 3y^{2} + 5y - 3y - 5$
$= y(3y + 5) - 1(3y + 5)$
$= (3y + 5)(y - 1)$.
Combining these,the volume is $4k \times (3y + 5) \times (y - 1)$.
Therefore,the possible dimensions of the cuboid are $4k$,$(3y + 5)$,and $(y - 1)$ units.