Originally Posted by BimmerFan52

Eeeeeeewwwww. That is a nasty looking engine. I'll bet your air and cabin filters are a mess too.

Anyone having to work on that engine will be muttering under their breath, and it won't be praise for you.

The panel which goes under the engine is commonly referred to as a "belly pan" or "splash pan". BMW refers to is as the "front engine encapsulation" and it is shown as #5 in the link below.
RealOEM.com * BMW E60 545i Air duct

In the past some manufacturers did utilize pans underneath the engine to keep the engine cleaner but BMW's under body panels are much more sophisticated.

BMW began 15 years ago innovating under-chassis aerodynamics as key to exceptionally low drag coefficients and high speed stability through under-chassis airflow control.

If you get a chance to look at the underside of your car on a lift you will see that BMW employs a number of panels all the way to the rear of the car to "smooth" the underside in order to create a controlled stream of air. This dramatically reduces drag by reducing air tumbling, which interrupts smooth flow and creates unwanted lift. We have all probably driven a car at high speed down the highway when the front end starts to feel light like it is being lifted. It is. Air is tumbling under the car and causing aerodynamic lift of the front end.

Smooth airflow under the car, coupled with control of the airflow into and out of the engine compartment, actually keeps the engine cooler than running without the belly pan.

If you are missing your belly pan I would replace it.

As for taking care of that mess under the hood. DO NOT USE A HOSE. Lots of electronics under the hood do not like to be flooded. Only you can decide how far you want to go in detailing, but at a minimum the major surface dirt should be removed.

The best approach is two pails and three rags. Put a SMALL amount of soap in the water of one pail, and just plain water in the second pail.

Begin by cleaning a small area with the soapy water, using the rags damp, but not dripping. Again, the idea is to not put a lot of water on the engine electronics. Swirl the rag several times in the soapy water as you clean that small area. Then using the second rag in the plain water wipe that small area, rinsing the rag several times. Then after the cleaned and rinsed area is beginning to dry, buff it dry with the third towel.

Rinse and refill both pails often and you should be able to get through it in no time. Once you have it back under control maintenance will become 10 minutes every few months with just a rag damp with water.

Mechanics will thank you, the engine will run cooler and you will have a much easier time identifying and catching leaks early.