<s>[INST] How to find the value of this expression Suppose I have a general quadratic equation $ ax^2 + bx + c = 0 $ 
And I have $ \alpha , \beta $ as roots, then what is the simplest way to find $ (a\alpha + b)^{-2} + (a\beta + b)^{-2} $ ?
I know the formula for finding the sum and product of roots, but that doesn't helped me in finding the value.
Any good hint is welcomed.
 [/INST] Some assumptions:
$c, \alpha, \beta \neq 0$ (the answer will be a bit modified without these assumptions)
Since $\alpha$ is a root therefore
\begin{align*}
a\alpha^2+b\alpha+c &=0\\
a\alpha+b & = \frac{-c}{\alpha}\\
\frac{1}{(a\alpha+b)^2} & = \frac{\alpha^2}{c^2}
\end{align*}
Thus
\begin{align*}
\frac{1}{(a\alpha+b)^2} + \frac{1}{(a\beta+b)^2}& = \frac{\alpha^2+\beta^2}{c^2}\\
&= \frac{(\alpha+\beta)^2-2\alpha\beta}{c^2}
\end{align*}
</s>