To factor the expression \( 12ab^3 + 20b \) completely, we start by identifying the greatest common factor (GCF) of the two terms.
- Identify the coefficients: The coefficients are 12 and 20. The GCF of 12 and 20 is 4.
- Identify the variable factors: The terms have \( b \) in common. The smallest power of \( b \) present in both terms is \( b^1 \).
Combining these, the GCF of \( 12ab^3 \) and \( 20b \) is \( 4b \).
Now, we can factor out \( 4b \) from the expression:
\[ 12ab^3 + 20b = 4b(3ab^2 + 5) \]
Next, we check if \( 3ab^2 + 5 \) can be factored further. Since \( 3ab^2 + 5 \) does not have a common factor and cannot be factored using any simple methods (it is a sum of terms with no common variables), it is already in its simplest form.
Thus, the completely factored form of the expression \( 12ab^3 + 20b \) is:
\[ \boxed{4b(3ab^2 + 5)} \]
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