p(2) = 0 => a*p(2)=0
p(2) = 0 and q(2) = 0 => p(2)+q(2) = 0
p(x) = 0 is in H
p and q are arbitrary elements of H. What you are showing is that H is closed under addition and multiplication by a scalar and is non-empty ie.
p is an element of H and q is an element of H => p+q is an element of H
and
p is an element of H => a*p is an element of H for an arbitrary scalar 'a'
and
there exists p such that p is an element of H.
This is the very basics of subspaces, so I suggest you talk to your teacher if you don't recognise what I am saying.