Short Answer 
The expression 8q^6r^3 + 27s^6t^3 is factored as a sum of cubes using the formula a^3 + b^3 = (a + b)(a^2 – ab + b^2). The final factorization results in (2q^2r + 3s^2t)(4q^4r^2 – 6q^2rs^2t + 9s^4t^2).
Step 1: Recognize the Sum of Two Cubes
The given expression, 8q^6r^3 + 27s^6t^3, can be identified as a sum of two cubes. To facilitate its factorization, we can express it in the standard form of cubes:
- First cube: (2q^2r)^3
- Second cube: (3s^2t)^3
This transformation prepares us for applying the cube identity.
Step 2: Apply the Sum of Cubes Formula
We will utilize the formula for factoring a sum of cubes, described as a^3 + b^3 = (a + b)(a^2 – ab + b^2). Here, let:
- a = 2q^2r
- b = 3s^2t
By substituting these values into the formula, we can set up the factorization.
Step 3: Complete the Factorization
Substituting a and b into the formula yields the following factorization:
- First part: (2q^2r + 3s^2t)
- Second part: (4q^4r^2 – 6q^2rs^2t + 9s^4t^2)
Thus, the final factorization of the original expression is (2q^2r + 3s^2t)(4q^4r^2 – 6q^2rs^2t + 9s^4t^2).