Let $\alpha ,\beta \in \mathbf{R},\beta \ne 0\text{ and }\alpha +i\beta$ be a root of $x^3+qx+r=0$ where, $q,r\in R.$ A cubic equation with real coefficients one of whose roots of $\alpha ,$is

As, $q,r\in \mathbf{R},\alpha -i\beta$ must be a root of $x^3+qx+r=0.$
Let the third root be $\gamma ,$ then

$\begin{array}{l}\alpha +i\beta +\alpha -i\beta +\gamma =0\\ \Rightarrow \gamma =-2\alpha \end{array}$

As $\gamma$ is a root of $x^3+qx+r=0,-8{\alpha }^3-2q\alpha +r=0$

$\therefore$ required equation is $8x^3 + 2qx – r = 0.$