Let $\alpha$ and $\beta$ be the roots of $x^2 - 2x - 1 = 0.$ Let $p(x)$ be a cubic polynomial such that \begin{align*} p(\alpha) &= 2 \alpha, \\ p(\beta) &= 2 \beta, \\ p(\alpha + \beta) &= 2 \alpha + 2 \beta, \\ p(\alpha \beta) &= 4. \end{align*}Find $p(x).$
