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Try to use full_simplify on the expression

sage: var('n')
n
sage: implicit_multiplication(True)
sage: F = (n - 1)^2 2^n n n^(n - 2) - (n - 1)2^(n - 1)n^(n - 1) - 2(n - 1)2^(n - 2)n^(n - 1) - 2(n^2 - 3n + 2)2^(n - 1)n^(n - 1)
sage: F.full_simplify()
0