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The extending structures problem for strict Lie 2-algebras is studied. To provide the theoretical answer to this problem, this paper introduces the unified product of a given strict Lie 2-algebra <inline-formula> <math display="inline"> <semantics> <mi mathvariant="fraktur">g</mi> </semantics> </math> </inline-formula> and 2-vector space <i>V</i>. The unified product includes some interesting products such as semi-direct product, crossed product, and bicrossed product. The paper focuses on crossed and bicrossed products, which give the answer to the extension problem and factorization problem, respectively.