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

Volume of a cuboid $=($ Length $) \times($ Breadth $) \times($ Height $)$

Volume $=12 ky ^{2}+8 ky -20 k$

We have $12 ky ^{2}+8 ky -20 k =4\left[3 ky ^{2}+2 ky -5 k \right]=4\left[ k \left(3 y ^{2}+2 y -5\right)\right]$

$=4 \times k \times\left(3 y ^{2}+2 y -5\right)$

$=4 k \left[3 y ^{2}-3 y +5 y -5\right]$              (Splitting the middle term)

$=4 k [3 y ( y -1)+5( y -1)]$ $=4 k[(3 y+5)(y-1)]$

$=4 k \times(3 y +5) \times( y -1)$

Thus, the possible dimensions are: $4 k ,\,(3 y +5)$ and $( y -1)$ units.