To factor the expression \( 48x^3 + 8x^4 \) completely, we can start by identifying the greatest common factor (GCF) in both terms.
The GCF of \( 48x^3 \) and \( 8x^4 \) is \( 8x^3 \).
Now, we can factor out \( 8x^3 \):
\[ 48x^3 + 8x^4 = 8x^3(6 + x) \]
Thus, the completely factored form of the expression is:
\[ \boxed{8x^3(6 + x)} \]