<s>[INST] Value of $ \frac{a^2 + b^2 + c^2}{ac^2 - ab^2} $ If  $a + b + c = 0$ , then value of $ \frac{a^2 + b^2 + c^2}{ac^2 - ab^2} $ is?
According to me if $ a + b + c = 0$, then $a + b = -c$

This implies $ (a + b)^2 = c^2$
This implies $a^2 + b^2 = c^2 - 2ab$

Putting this value in  $ \frac{a^2 + b^2 + c^2}{ac^2 - ab^2} $ we get

$ \frac {c^2 - 2ab + c^2}{ac ^ 2 - ab}  = \frac {2c^2 - 2ab}{a(c^2 - b)} $
But I'm stuck after this. Am I right in the above steps, if yes, then how should I continue?

 [/INST] Put $a=-(b+c)$ we have  after   simplification 
$$
\frac{a^2+b^2+c^2}{ac^2-ab^2}=2\,{\frac {{c}^{2}+cb+{b}^{2}}{ \left( c+b \right)  \left( -{c}^{2}+{b
}^{2} \right) }}=2 \frac{b^3-c^3}{(b^2-c^2)^2}
$$
</s>