If you take P(x) and subtract a, you must have roots at 1,3,5,7 thus it must be of the form:

P(x)-a = (x-1)(x-3)(x-5)(x-7)Z(x)

additionally you can use the same logic to get the bottom ones:

P(x)+a = (x-2)(x-4)(x-6)(x-8)Y(x)

But then adding these you can get P(x) as:

P(x)=1/2((x-1)(x-3)(x-5)(x-7)Z(x)+(x-2)(x-4)(x-6)(x-8)Y(x)),

To make this small but have integer coefficients Y(X) and Z(x) must be a multiple of 2, so:

P(x)=(x-1)(x-3)(x-5)(x-7)z(x)+(x-2)(x-4)(x-6)(x-8)y(x)

therefore a must me a multiple of (1-2)(1-4)(1-6)(1-8) and all the other values plugged into this equation when z(x)=1 and y(x)=1, so the smallest it can be is 3*3*5*7=315 (B) EDITED at 13:38