Introduction:
To find the possible expressions for the dimensions of a cuboid given its volume, we need to consider the factors of the volume expression. In this case, the volume of the cuboid is given as 12ky^2 - 8ky - 20k.

Factors of the Volume Expression:
To determine the possible expressions for the dimensions, we need to factorize the volume expression and analyze its factors. Let's factorize the given volume expression:

12ky^2 - 8ky - 20k

Step 1: Factor out the common factor '4k':
4k(3y^2 - 2y - 5)

Step 2: Factorize the quadratic expression inside the parentheses:
4k(3y + 1)(y - 5)

Dimensions of the Cuboid:
Now, we have the factored expression 4k(3y + 1)(y - 5). From this, we can determine the possible expressions for the dimensions of the cuboid.

Length:
The length of the cuboid can be expressed as any factor of (3y + 1) multiplied by 4k. Therefore, the possible expressions for the length are:
- 4k
- (3y + 1)
- (4k)(3y + 1)
- (4k)(y - 5)

Width:
The width of the cuboid can be expressed as any factor of (y - 5) multiplied by 4k. Therefore, the possible expressions for the width are:
- 4k
- (y - 5)
- (4k)(3y + 1)
- (4k)(y - 5)

Height:
The height of the cuboid can be expressed as any factor of 4k. Therefore, the possible expressions for the height are:
- 4k
- (3y + 1)
- (y - 5)

Conclusion:
The possible expressions for the dimensions of the cuboid, given its volume of 12ky^2 - 8ky - 20k, are:
- Length: 4k, (3y + 1), (4k)(3y + 1), (4k)(y - 5)
- Width: 4k, (y - 5), (4k)(3y + 1), (4k)(y - 5)
- Height: 4k, (3y + 1), (y - 5)

These expressions represent the different combinations of dimensions that can result in the given volume of the cuboid.