To find the product of \( 6(4z^2 + 2z + 1) \), you need to distribute the \( 6 \) to each term inside the parentheses.
Starting with the expression:
\[ 6(4z^2 + 2z + 1) \]
Distributing \( 6 \):
- \( 6 \times 4z^2 = 24z^2 \)
- \( 6 \times 2z = 12z \)
- \( 6 \times 1 = 6 \)
Now, combine all the results:
\[ 24z^2 + 12z + 6 \]
Thus, the simplified product is:
\[ \boxed{24z^2 + 12z + 6} \]