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For each choice of critical point \\gamma of f(x), we consider the family\ng_{\\gamma,m}(x)= (x - \\gamma)^2 + m + \\gamma, m \\in K.\nFor fixed n \\geq 3 and nearly all values of \\gamma, we show that there are only finitely many m such that g_{\\gamma,m} has a newly reducible nth iterate. For n = 2 we show a similar result for a much more restricted set of \\gamma. 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