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 \(P(x)\)?

(第二十五届AIME1 2007 第8题)