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)