Let X = $\{p \in P\_2(\mathbb{R}): p'(-1)-2 \int\_0^1 p(x) dx = 0\}$

I need to find a basis for X and demonstrate that it is a basis. So far I have deduced that the polynomials in the set must satisfy $c = (-4/3)a$ , so they must be of the form $p(x) = ax^2 + bx + (-4/3)a$.

How do I determine the vectors in the basis X working from the standard basis for $P^2$ of $\{1,x,x^2\}$?