Determine whether W is a subspace of V

my solution:

First step: W is nonempty because it contains the zero polynomial (a=b=c=0)

Second step: Let

and

Then

So p(x) + q(x) is also in W (because it has the right form). Similarly, if k is a scalar, then

so kp(x) is in W.

Thus, W is a nonempty subset of that is closed under addition and scalar multiplication. therefore, W is a subspace of

Was just wondering if this was all good, becuase the initial condition of the set stated above (a*b*c=0) confused me a bit