Cubic cyclic field families

Question

Let $\mathbb{Q}$ is the field of rational numbers. If $f(t,x)=x^3+(t+2)x^2+(t-1)x-1$, show that for any integer $t$, adjoining a root $r$ of $f(t,x)$ to $\mathbb{Q}$ will yield a cyclic cubic field. Once this is shown, if possible, find a quintic (degree $5$) polynomial $f(t,x)$ (coefficients in terms of $t$) such that for any integer $t$, adjoining a root of $f(t,x)$ to $\mathbb{Q}$ will yield a cyclic quintic field. Thanks for help.

Answer 1

Note that if (for fixed $t$) $\alpha$ is a root of $f(t, x)$ then so is $-1-\alpha^{-1}$. The transformation $y \mapsto -1 -y^{-1}$ has order three. Maybe it is possible to find something similar for the quintic case.