Solve the following equations:
\begin{align}\left\{ \begin{array}{c} x_1+x_2+x_3+x_4=6 \\ x_1{}^2+x_2{}^2+x_3{}^2+x_4{}^2=10 \\ x_1{}^3+x_2{}^3+x_3{}^3+x_4{}^3=18 \\ x_1{}^4+x_2{}^4+x_3{}^4+x_4{}^4=34 \\ \end{array} \right.\end{align}
method1:
consider a polynomial $f(x)$ whose roots are $x_1, x_2,x_3,x_4$
\begin{align}f(x)=x^4+c_1 x^3+c_2 x^2+c_3 x+c_4\end{align}
then $c_1=-6$, $c_2=13$, I'm stuck at how to obtain $c_3$ and $c_4$, ie. $x_1x_2 x_3+ x_1x_2x_4+x_1x_3x_4+x_2x_3x_4$ and $x_1x_2 x_3 x_4$
Are there any other methods? tips?