To factor the expression $6x(3x + 1)^3 - (3x + 1)^4$, let's follow these steps:

Step 1: Identify the common factor

The expression can be rewritten as: $6x(3x + 1)^3 - 1 \cdot (3x + 1)^4$

Here, $(3x + 1)^3$ is a common factor. Let's factor it out: $(3x + 1)^3 \cdot \left[ 6x - (3x + 1) \right]$

Step 2: Simplify the expression in the brackets

Simplify the expression inside the brackets: $6x - (3x + 1) = 6x - 3x - 1 = 3x - 1$

Step 3: Write the fully factored form

Now, substituting back, the factored form of the expression is: $(3x + 1)^3 \cdot (3x - 1)$

Thus, the factored form of the expression $6x(3x + 1)^3 - (3x + 1)^4$ is: $\boxed{(3x + 1)^3 \cdot (3x - 1)}$

Would you like more details, or do you have any questions?

Here are some related questions to explore:

1. How do you factor expressions with different powers?
2. What are some common techniques for factoring polynomials?
3. Can all polynomials be factored using simple techniques?
4. What is the significance of factoring in solving polynomial equations?
5. How can you verify that your factored form is correct?

Tip: When factoring, always check for the greatest common factor (GCF) first, as it simplifies the process.