The equation whose roots are the A.M and twice the H.M. between the roots of the equation x² + ax –b = 0 is

The given equation is $x^2+ax-b=0$

Let $\alpha ,\beta$ are the roots of the equation 

We know that $\alpha +\beta =-a$ and $\alpha \beta =-b$

The Arithmetic means of the roots is $\frac{\alpha +\beta }{2}=-\frac{a}{2}$ and the twice theHormonic mean between $\alpha ,\beta$ is $\frac{4\alpha \beta }{\alpha +\beta }=\frac{4b}{a}$ 

The required quadratic equation is ${\mathrm{x}}^2+\left(\frac{\mathrm{a}}{2}-\frac{4\mathrm{b}}{\mathrm{a}}\right)\mathrm{x}-2\mathrm{b}=0$

simplifying the above equation $\boxed{2a{x}^2+\left(a^2-8b\right)x-4ab=0}$