We know that, in a cubic equation with rational coefficients has a irrational root $\alpha +\sqrt{\beta }$, where $\alpha$ and $\beta$ are rational and $\beta$ is not a perfect square, then it has also a conjugate root $\alpha -\sqrt{\beta }$.

Hence, $1, 2+\sqrt{5}, 2-\sqrt{5}$are the roots of cubic equation.

We know that, if  $\alpha , \beta , \gamma$ are the roots of cubic equation, then its equation is ${\mathrm{x}}^3-\left(\alpha +\beta +\gamma \right){\mathrm{x}}^2+\left(\alpha \beta +\alpha \gamma +\beta \gamma \right)\mathrm{x}-\left(\alpha \beta \gamma \right)$.

Let, $\alpha =1, \beta =2+\sqrt{5}, \gamma =2-\sqrt{5}$

$\alpha +\beta +\gamma =1+2+\sqrt{5}+2-\sqrt{5}=5$, $\alpha \beta +\alpha \gamma +\beta \gamma ==1\left(2+\sqrt{5}\right)+\left(2+\sqrt{5}\right)\left(2-\sqrt{5}\right)+1\left(2-\sqrt{5}\right)=2+\sqrt{5}+4-5+2-\sqrt{5}=3$

$\alpha \beta \gamma =1\left(2+\sqrt{5}\right)\left(2-\sqrt{5}\right)=-1$.

Hence, required cubic equation is $x^3-5x^2+3x+1=0$.