The P versus NP problem is a cornerstone of theoretical computer science, asking whether problems that are easy to check are also easy to solve. "Easy" here means solvable in polynomial time, where the computation time grows proportionally to the input size. While this problem's origins can be traced to John Nash's 1955 letter, its formalization is credited to Stephen Cook and Leonid Levin. Despite decades of research, a definitive answer remains elusive. Central to this question is the concept of NP-completeness. If even one NP-complete problem could be solved efficiently, it would imply that all problems in NP could be solved efficiently, proving P equals NP. This research proposes that a notoriously difficult NP-complete problem can be solved efficiently, thereby potentially establishing the equivalence of P and NP. This work is an expansion and refinement of the article "Note for the P versus NP problem", published in IPI Letters.