Let v \in V. Then v is a polynomial of degree less than or equal to 2 with v(0)=0. Expressed this way v = a+bx+cx^2 where a,b,c are in the field over which V is a vector space. As you said this implies a = 0, so v = bx+cx^2. Therefore v (an arbitrary element of V) can be written as a linear combination of elements in {x,x^2} therefore {x,x^2} is a spanning set. Show this set is also linearly independent (by assuming ax + bx^2 = 0 and concluding that a = b = 0) and you've got yourself a basis.

Or show that {x, x^2} is the minimal spanning set; that is removing any element destroys the spanning property.