As Daniel noted, P1 and P2 must have a common root. That root can be found by calculating 2*P1(x) - P2(x) = 2*(x^2 + (k-29)x - k) -(2x^2+(2k-43)x + k) = -15x - 3k = -15*(x+k/5).
x = -k/5 is the common root of P1 and P2. Plug that in and equate the equations to zero: (-k/5)^2 + (k-29)*(-k/5) - k = 0 and 2*(-k/5)^2 + (2k-43)*(-k/5) + k = 0.
The two equations reduce to (-4/25)k^2 + (24/5)k = 0 and (-8/25)k^2 + (48/5)k = 0 respectively. These are the same quadratic, just one has coefficients twice as large. The roots are k=30 and k=0.
Then the total set of k such that P1 and P2 are factors of the cubic P(x) are {0,30}. The maximum value of k must be 30.
Another approach
