Let $ r $ be the common root, so \begin{align*} r^2 + ar + 1 &= 0, \\ r^2 - r - a &= 0.\end{align*}Subtracting these equations, we get $ ar + r + a + 1 = 0 $. This factors as $ (r + 1)(a + 1) = 0, $ so $ r = -1 $ or $ a = -1 $. If $ r = -1, $ then $ 1 - a + 1 = 0, $ so $ a = 2 $. If $ a = -1, $ then $ x^2 - x + 1 = 0, $ which has no real roots. So the only possible value of $ a $ is $ \boxed{2} $.