The polynomial $P(x)$ is cubic . What is the largest value of $k$ for which the polynomials $Q_1(x) = x^2 + (k-29)x - k$ and $Q_2(x) = 2x^2+ (2k-43)x + k$ are both factors of $P(x)$ ?

Let's think step by step and output the final answer within \boxed{}.
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