Here is the query
from a in this._addresses
where a.Street.Contains(street) || a.StreetAdditional.Contains(streetAdditional)
select a).ToList(
The most obvious one:
from a in this._addresses
where (a.Street != null && a.Street.Contains(street)) || (a.StreetAdditional != null && a.StreetAdditional.Contains(streetAdditional))
select a).ToList()
Alternatively you could write an extension method for Contains that accepts a null argument without error. Some might say that it is not so pretty to have such a method, because it looks like a normal method call, but is allowed for null values (thereby setting aside normal null-checking practices).