A warrant requires probable cause. A warrant is required to search but the courts have carved out exceptions to the warrant requirements.
Your question is too vague but any search can be questioned and challenged..
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You can always object at the scene.
Police have to make a thousand judgement calls a day. Sometimes they goon it up.
If police find something you can later challenge that search.
Your prior clean history has no relevance to an officer's probable cause.
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I don't know if this is specifically related to an actual situation or just in the hypothetical. If it's an actual case, then your attorney can examine all the evidence to determine whether the search was legal or not.
If this is just a general question...
In order to search, the police must have a valid search warrant or a recognized exception to the need for a warrant. They can search with consent and if that's the case, they don't even need probable cause. If they are going to justify a warrantless, nonconsensual search, they must develop probable cause. Yes, it is subjective, but it can always be challenged in court.
Probable cause is a set of circumstances and facts that give the officer a strong suspicion or reasonable likelihood that the crime-related evidence is going to be found in the area they wish to search. Just developing that probable cause doesn't automatically mean they can just barge in and search. In certain circumstances, they must still obtain a warrant. Other "exigent circumstances" may excuse the need for a warrant, but the probable cause must still exist.
The above answer is for general information only and is based on the information you posted. Every case is fact dependent, so to get a thorough analysis of your situation, you will need to consult face to face with an attorney licensed to practice in the jurisdiction where the incident took place. Do not conclusively rely on any information posted online when deciding what to do about your case.
If the police are going to search your home they need probable cause and a warrant (or an exception to the warrant requirement.) The car and the home are not treated the same in the law or practice. That's why the police ask permission: because that is an exception to the warrant requirement. As with all legal questions, lots of facts are necessary for a complete and correct answer.
Probable cause is actually not subjective. The main case is United States v. Whren.