The equation of lowest degree with rational coefficients having a root $\sqrt{5}+\sqrt{2}i$  is

$\alpha =\sqrt{5}+\sqrt{2}i,\beta =\sqrt{5}-\sqrt{2}i$

$\gamma =-\sqrt{5}+\sqrt{2}i,\delta =-\sqrt{5}-\sqrt{2}i$

$\begin{array}{l}\Rightarrow (x^2-x(\alpha +\beta )+\alpha \beta )(x^2-x(\gamma +\delta )+\gamma \delta )=0\\ \Rightarrow (x^2-x\left(2\sqrt{5}\right)+7)(x^2+2\sqrt{5}x+7)=0\\ \Rightarrow {(x^2+7)}^2-{\left(x2\sqrt{5}\right)}^2=0\\ \Rightarrow x^4-6x^2+49=0\end{array}$