To find the total number of songs Jonah has recorded after \( w \) weeks, we can start with the number of songs he already has, which is 32. Each week, he records 6 new songs, so after \( w \) weeks, he will have recorded an additional \( 6w \) songs.
The expression for the total number of songs recorded after \( w \) weeks is:
\[ 32 + 6w \]
This expression accounts for the initial 32 songs plus the 6 songs recorded each week for \( w \) weeks.