$\alpha,\beta,\gamma$ are roots of cubic equation $x^3+4x-1=0$ If $\alpha,\beta,\gamma$ are the roots of the equation $x^3+4x-1=0$ and $\displaystyle \frac{1}{\alpha+1},\frac{1}{\beta+1},\frac{1}{\gamma+1}$ are the roots of the equation 
$\displaystyle 6x^3-7x^2+3x-1=0$. Then value of $\displaystyle \frac{(\beta+1)(\gamma+1)}{\alpha^2}+\frac{(\gamma+1)(\alpha+1)}{\beta^2}+\frac{(\alpha+1)(\beta+1)}{\gamma^2}=$
$\bf{My\; Try}::$ Given $x=\alpha,\beta,\gamma$ are the roots of the equation $x^3+4x-1=0$, Then 
$x^3+4x-1 = (x-\alpha)\cdot(x-\beta)\cdot (x-\gamma)$
put $x=-1$, we get $(1+\alpha)(1+\beta)(1+\gamma) = 6$
Now $\displaystyle \frac{(\beta+1)(\gamma+1)}{\alpha^2}+\frac{(\gamma+1)(\alpha+1)}{\beta^2}+\frac{(\alpha+1)(\beta+1)}{\gamma^2} = 6\left\{\frac{1}{(\alpha+1)\alpha^2}+\frac{1}{(\beta+1)\beta^2}+\frac{1}{(\gamma+1)\gamma^2}\right\}$
Now how can I solve after that,
Help required
thanks

Hint: Since $x^{3} + 4x - 1 = (x-\alpha)(x-\beta)(x-\gamma)$, we note that, by the theory of symmetric polynomials, 
$$0 = \alpha+\beta+\gamma = s_{1}(\alpha, \beta, \gamma)$$
$$4 = \alpha\beta + \beta\gamma+\alpha\gamma = s_{2}(\alpha, \beta, \gamma)$$
$$1 = \alpha\beta\gamma = s_{3}(\alpha, \beta, \gamma)$$
As suggested by Mariano Suárez-Alvarez, we can observe immediately that the expression
$\displaystyle \frac{(\beta+1)(\gamma+1)}{\alpha^2}+\frac{(\gamma+1)(\alpha+1)}{\beta^2}+\frac{(\alpha+1)(\beta+1)}{\gamma^2}$
is symmetric in $\alpha, \beta, \gamma$, i.e. it is invariant under permuting $\alpha, \beta, \gamma$. Therefore, I would put everything under a common denominator. The numerator then becomes
$$\beta^{2}\gamma^{2}(\beta\gamma+\gamma+\beta+1) + \alpha^{2}\gamma^{2}(\alpha\gamma+\gamma+\alpha+1) + \alpha^{2}\beta^{2}(\alpha\beta + \alpha + \beta+1)$$
It can easily be seen that this is a symmetric polynomial in $\alpha, \beta, \gamma$ as well. It can therefore be written in terms of $s_{1}, s_{2}, s_{3}$. The denominator is $\alpha^{2}\beta^{2}\gamma^{2} = s_{3}^{2} = 1^{2} = 1$. If you would like more information on how to write the numerator in terms of the symmetric polynomials, please comment and I'd be happy to add information.  
