<s>[INST] Where is the mistake in my reasoning? I have a statement that says:

If $a^2 + b^2 + c^2 = 2$ and $(a + b + c)(1 + ab + bc + ac) = 3^2$
What is the value of $( a + b + c )$ ?

My reasoning was:
$a^2 + b^2 + c^2 = 2$, rewritten as:


*

*$(a + b + c)^2 = 2 + 2ab + 2ac + 2bc$


Since,  $(a + b + c)(1 + ab + bc + ac) = 3^2$


*$(1 + ab + bc + ac) = \frac{9}{(a + b + c)}$


Now, replacing in the 1. 


*Factorize $(a + b + c)^2 = 2(1 + ab + ac + bc)$

*Replacing $(a + b + c)^2 = \frac{18}{a + b + c}$

*Multiplying by $(a + b + c)$

*$(a + b + c)^3 = 18$. Thus $a + b + c = \sqrt[3]{18}$. That is my result.
But, the correct result should be $4$, then where is my mistake ?
 [/INST] Suppose $a^2+b^2+c^2=2$ and $(a+b+c)(1+ab+bc+ac)=k$. Then, setting $s=a+b+c$ and $q=ab+bc+ac$, we get
$$
s^2=2+2q=2(1+q),\qquad s(1+q)=k
$$
Therefore $1+q=s^2/2$ and
$$
s^3=2k
$$
so that $s=\sqrt[3]{2k}$. If $s$ has to be $4$, then $2k=64$, that is, $k=32$.
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