This shoudn't be too hard of a problem..

1.) If a and b are in A,then a-b is in A.

Proof: Since a and b are in A, then a(2) =b(2)=0. Because we are in Z[x] this implies that 2 is a root and they are divisible by (x-2). So we can write them in the form a = (x-2)a', b = (x-2)b', where a' and b' are in Z[x]. Then factoring the (x-2) out we see a-b = (x-2)(a'-b'). This means a-b is divisible by x-2 and has a root of 2, so a-b is in A.

2.) ar is in A for a in A and r in Z[x]

Similar to above, we see that a = (x-2)a', and ar = (x-2)(a')(r) which is divisible by x-2 and therefore has a root at 2, so ar is in A for a in A and r in Z[x].