Given equation is

$2x\left(2x+1\right)=1\Rightarrow 4x^2+2x-1=0$ .........(1)

roots of equation $\left(1\right)$

are $\alpha$ and $\beta$

$\therefore \alpha +\beta =-\frac{1}{2}$

$\Rightarrow \beta =-\frac{1}{2}-\alpha$   ........(2)

and

$4{\alpha }^2+2\alpha -1=0\Rightarrow {\alpha }^2=\frac{1}{4}-\frac{\alpha }{2}$ .........(3)

now

$-2\alpha \left(\alpha +1\right)=-2{\alpha }^2-2a$

$=-2\left(\frac{1}{4}-\frac{\alpha }{2}\right)-2\alpha =-\frac{1}{2}-\alpha =\beta$