If $\alpha, \beta,\gamma$ be the roots of $2x^3+x^2+x+1=0$, show that..........?

Notice that if we multiplly the polynomial with $x-1$ we get $$x^3(x-1)+(x-1)(x^3+x^2+x+1)=0$$ so $$2x^4-x^3-1=0\implies \boxed{{1\over x^3}= 2x-1}$$

So your expression is $$E=(2\beta + 2\gamma -2\alpha -1)(2\beta -2\gamma +2\alpha -1)(-2\beta + 2\gamma +2\alpha -1)$$

Now notice that $$\beta + \gamma +\alpha =-{1\over 2}$$ so $$E = -8(2\alpha +1)(2\beta +1)(2\gamma +1)=$$

$$=-64 (\alpha +{1\over 2})(\beta +{1\over 2})(\gamma +{1\over 2}) $$ $$=-32\cdot p(-{1\over 2})=-16$$