To factor the expression $(x-5)(x+4)^2-(x-5)^2(x+4)$, we can follow these steps:

1. Factor out the common factor:
   The common factor between the two terms is $(x-5)(x+4)$.

   $(x-5)(x+4)^2 - (x-5)^2(x+4) = (x-5)(x+4)\left[(x+4) - (x-5)\right]$

2. Simplify the expression inside the brackets: $(x+4) - (x-5) = x + 4 - x + 5 = 9$

3. Multiply the common factor by the simplified expression: $(x-5)(x+4) \cdot 9 = 9(x-5)(x+4)$

So, the fully factored form of the expression is: $\boxed{9(x-5)(x+4)}$

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:

1. How would you factor the expression $(x-3)(x+2)^2 - (x-3)^2(x+2)$?
2. What are the roots of the equation $9(x-5)(x+4) = 0$?
3. How does the distributive property apply when factoring expressions like this?
4. Can this method be used for any polynomial expression with common factors?
5. How can we expand the factored form $9(x-5)(x+4)$ back into its original form?

Tip: When factoring, always check for common factors first, as they can greatly simplify the process.