If `alpha,beta,gamma` are the roots of the equation `x^3 + 4x +1=0` then `(alpha+beta)^(-1)+(beta+gamma)^(-1)+(gamma+alpha)^(-1)=`

Explanation

For the given equation `alpha + beta + gamma = 0`
`alpha beta + beta gamma + alpha gamma = 4, alpha beta gamma = -1`
Now, `(alpha + beta)^(-1)+(beta + gamma)^(-1) + (gamma + alpha)^(-1).=(-gamma)^(-1)+(-alpha)^(-1) +(-beta)^(-1)`
`=(alpha beta + beta gamma + alpha gamma)/(alpha beta gamma)`
= `-(4)/(-1)`
= 4.