Relation between roots and coefficients - manipulation of identities 
The polynomial $x^3+3x^2-2x+1$ has roots $\alpha, \beta, \gamma$ . Find $$\alpha^2(\beta + \gamma) + \beta^2(\alpha + \gamma) + \gamma^2(\alpha + \beta)$$

I tried finding the relation using $-b/a$, $c/a$ and $-d/a$. I couldn’t seem to find anything. I also tried solving for one root but it gave me back the polynomial but with the root as the variable. Also the polynomial can not be factorised.

$a,b,c$ are the three roots.
$$
\begin{align}
&a^2*(b+c)+b^2*(a+c)+c^2*(a+b)\\
={}&(a+b+c)*(a^2+b^2+c^2)-(a^3+b^3+c^3)\\
={}&(-3)*(a^2+b^2+c^2)-(a+b+c)^3\\
={}&(-3)*((a+b+c)^2-2ab-2ac-2bc)-(a+b+c) * (a^2+b^2+c^2) + ab(a+b) + ac(a+c) + bc(b+c)\\
={}& (-3)*(9-2*(-2))-(-3)*(9-2*(-2)) + ab(a+b+c-c) + ac(a+b+c-b) + bc(a+b+c-a)\\
={}&(a+b+c)(ab+ac+bc)-3abc\\
={}&(-3)*(-2)-3*(-1)\\
={}&6-(-3)\\
={}&9
\end{align}
$$
