Let's factor the expression $a^5 - 4ab^4$.

Given expression: $a^5 - 4ab^4$

Step 1: Factor out the greatest common factor (GCF).

The GCF of the terms $a^5$ and $4ab^4$ is $a$. So, factor $a$ out:

$a(a^4 - 4b^4)$

Step 2: Notice that $a^4 - 4b^4$ is a difference of squares.

The difference of squares can be factored as: $a^4 - 4b^4 = (a^2 - 2b^2)(a^2 + 2b^2)$

Step 3: Substitute this back into the factored expression:

$a^5 - 4ab^4 = a(a^2 - 2b^2)(a^2 + 2b^2)$

So the fully factored form of $a^5 - 4ab^4$ is:

$\boxed{a(a^2 - 2b^2)(a^2 + 2b^2)}$

Would you like more details or have any questions?

Here are 5 related questions to consider:

1. How would you factor the expression $a^6 - b^6$?
2. What would be the factorization of $a^4 - 16b^4$?
3. Can you apply the difference of squares method to factor $x^4 - 16$?
4. How do you recognize and factor a perfect square trinomial?
5. What happens when you try to factor $a^5 + 4ab^4$?

Tip: When dealing with polynomial expressions, always start by factoring out the greatest common factor (GCF) before looking for other factorization methods.