Minimise $|z_az_b + z_cz_d|$ where $\{z\}$ are the roots of $x^4 + 14x^3 + 52x^2 + 56x + 16$ Let $f(x) = x^4 + 14x^3 + 52x^2 + 56x + 16.$ Let  $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_az_b + z_cz_d|$ where ${a, b, c, d} = {1, 2, 3, 4}.$
So I have tried Vieta, but it is try too complex and plugging in a simple value to try to find a root failed, so I think this is going to be much harder than finding the four roots-- I wonder if there are more involved techniques to solve this problem.

I claim the minimum is $8$. I will use dxiv suggestion. Let $g(y) = y^4+7y^3+13y^2+7y+1$ Obviously all roots are real and negative since we have 
\begin{array}{cccccc}
  y & -5 & -2 & -1 & -0.5 & 0 \\
  g(y) & + & - & + & - & + \\
\end{array}
Let $a,b,{1\over a},{1\over b}$ be all roots for $g$ where $a,b,$ are negative.
So we have next possible expresions
\begin{eqnarray}
E_1 &=&ab+{1\over ab} >2\\ 
E_2 &=&{a\over b}+{b\over a} >2\\ 
E_3 &=&a{1\over a}+b{1\over b} =2\\ 
\end{eqnarray}
and does are all. So $E_{\min}=2$. So the minimum value for the given expression is $4\cdot 2 =8$
