<s>[INST] Finding solutions to a system of equations using elementary symmetric polynomials Find the value of a, b, and c given:
$a^2 + b^2 + c^2 = 129$
$ab + ac + bc = -4$
$(a^2)(b^2) + (a^2)(c^2) + (b^2)(c^2)$ = 984
I attempted this problem using elementary symmetrical polynomials and found that the second equation can be written as $e_2(a, b, c)$. However, the squares in the other equations made me quite confused and I was unable to write them in the previous from. If I was able to write it in the form of elementary symmetrical polynomials, I should have been able to use simple substitution to find the value of $e_1, e_2$, and $e_3$. This then would have allowed me to form a cubic equation, where I could find the solutions by solving it.
However, as I mentioned before, I'm a little bit stuck on the part where I need to find the values of $e_1, e_2, e_3$. Any help would be extremely appreciated :) Also, is there any other way to solve for a, b, and c using elementary symmetric polynomials and substitution, but not cubic equations?
Update:
Thanks to @Gerry_Myerson, I've worked out that the first equation is:
$(e_1)^2 - 2e_2 = 129$
and the second equation is:
$e_2 = -4$
However, I still don't know how to work out the last one.
 [/INST] \begin{align} 
a^2 + b^2 + c^2 &= 129
\tag{1}\label{1}
,\\
ab + bc + ac &= -4
\tag{2}\label{2}
,\\
a^2b^2 + b^2c^2 + a^2c^2&=984
\tag{3}\label{3}
.
\end{align} 
\begin{align}
\eqref{2}^2:\quad
a^2b^2+a^2c^2+b^2c^2+2abc(a+b+c)
&=16
,\\
abc(a+b+c)&=-484
,\\
\eqref{1}+2\cdot\eqref{2}:\quad
(a+b+c)^2&=121
.\\
a+b+c&=\pm 11
,\\
abc&=-\frac{484}{a+b+c}
=\mp 44
,
\end{align} 
and we have two cubics
\begin{align}
x^3-11x^2-4x+44
,\\
y^3+11y^2-4x-44
\end{align}
with roots $(-2, 2, 11)$
and $(-2, 2, -11)$.
</s>