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Good morning. Under Utah law a step parent can adopt providing the birth father has paid no support and has had no contact with the child for one year, then stepfather may adopt in most states without permission of the birth father. Before adopting a stepchild the adopting parent must be married to the adoptee's custodial parent; the adopting parent or the adopting parent's spouse must be at least 10 years older than the adoptee; and the adoptee must have lived with the custodial parent and the stepparent for one year (this can be waived by the judge).
If everyone who must consent to the adoption does so, and if everyone who has the right to notice of the adoption and the right to intervene waives those rights, then the procedures for adopting a stepchild are relatively simple:
You must file a petition for adoption the required documents, attend the hearing that is scheduled, and the judge will sign the Adoption Decree.
The court will not add his name to the birth certificate unless he is the biological father. I hope this helps with your question.