gonna throw my 2cents in here since i went down this road recently, i found it was best to exhaust out a window and it can be done fairly cheaply
your local home improvement store may sell inline fan kits in various sizes from 100mm or 4" and larger in the kitchen/shower diy sections should have various parts to customize this to your setup
the kits don't usually have an 'Inline Backdraft Damper' so id get one and have it at the window mount to stop backdrafts so you can still solder during the winter,
how you mount it to the window will be dependant on your window type, my house is from the 1950s and has those old school spring pull up windows so i got some wood, drilled a hole for the fan and the force from the window holds it firmly in place.
with hindsight i would've changed a few things with my setup as originally i bought a "Micron FUME Extractor Desk Swing Arm T1297" and thought running a tube from that to the window would be enough, in reality the fan was aweful and pushed bugger all air and i got a lot of backdraft so i had to rethink and ended up modding my setup with the inline fan which works way better and i wouldn't even need the micron fan except it already came with a 100mm duct mount on the back and the spring arm helps push/pull it toward or away from your workspace as needed
if i was to redo that part id used loc-line tubing or something like the exhuast with a hood on the aliexpress image9 below instead of the micron fan, the only issue being the largest size is 75mm so either you could get a 75mm inline fan and tubing for a more compact setup, or adapt 100mm to 75mm as 100mm inline fans are more common.
try to go with a mixflow inline fan as they should push more airflow than some cheaper ones, locally the cheapest inline fan was an axial type at 119 m3/hr noise of 40dB another i found on ebay was a mixflow and had two speed modes of 145m3/hr or 187m3/hr at 34dbA or quieter at lower speed and was only $20aud more than the cheaper axial type
Edit: another option for those not wanting to exhaust out a window and want a simpler setup, get a decent quality true HEPA air purifier and plonk it close to your soldering, personally i got an Arovec AV-P152PRO which has 250m3/hr on max, and was under $200aud on sale. at max fan speed it soaks up the fumes easily, the only downside of this method aside from being a bit bulky on the desk depending on the model you get, is you'll need to replace the air filters overtime which costs whereas a nice exhaust setup you don't need to worry about replacing filters