2007 AIME I Problem 8

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)?

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.