I am not sure from your answer exactly what is showing on the background search. If there is inaccurate information, go to the California Attorney General Website at www.ag.ca.gov and fill out a form available online to have your record corrected to show dismissal of the case. I would think that having the entire record removed is going to be an uphill battle. What dismissal means is that the prosecution failed to prove your guilt beyond a reasonable doubt, not that you were innocent. You certainly may have been innocent, 851.8 relief is not available once two years has passed.
Depending on the evidence, you can do a petition to seal and destroy the arrest record under Penal Code Section 851.8, but the standard of proof is high and the chance you would win is low. Otherwise, the record of the court will remain as is.
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Your biggest hurdle is that there is a 2 year statute of limitations from the date of dismissal on filing a motion for a finding of factual innocence. If such a motion were granted, your record would be sealed, which is a true expungement. Since you're past that date, your motion would be untimely and would be denied. Not sure what else you could do at this point.
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See an Attorney about an 851.8 petition, I do them often
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