vector spaces & subspaces

Determine whether W is a subspace of V

$\displaystyle V=P_2$

$\displaystyle W = \left\{ a + bx + cx^2 : abc=0 \right\}$

my solution:

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

Second step: Let

$\displaystyle p(x) = a + bx + cx^2$

and

$\displaystyle q(x) = d + ex + fx^2$

Then

$\displaystyle p(x) + q(x) = \left( a+d \right) + \left( b+e \right)x + \left( c + f \right)x^2 $

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

$\displaystyle kp(x) = ka + kbx + kcx^2 $

so kp(x) is in W.

Thus, W is a nonempty subset of $\displaystyle p_2$ that is closed under addition and scalar multiplication. therefore, W is a subspace of $\displaystyle p_2$

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