Question

$x^{2} + ax + 2 = 0\int_{\beta}^{\alpha}\mspace{2mu}\ \alpha + \beta = - a\ \alpha\beta = 2$

$${x^{2} + bc + 6 = 0\int_{\gamma}^{\beta}\mspace{2mu}\ \beta + \gamma = - b\ \beta\gamma = 6 }{x^{2} + cx + 3 = 0\bigwedge_{\alpha}^{\gamma}\mspace{2mu}\ \gamma + \alpha = - c\ \gamma\alpha = 3 }{\Rightarrow \ (\alpha\beta\gamma)^{2} = 36\ \Rightarrow \ \alpha\beta\gamma = \pm 6 }$$If $\alpha\beta\gamma = 6,\alpha = 1,\beta = 2,\gamma = 3,a + b + c = - 2(3 + 3) = - 12$

If $\alpha\beta\gamma = - 6,\alpha = - 1,\beta = - 2,\gamma = - 3,\ a + b + c = 12$