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 $\{a,b,c,d\}=\{1,2,3,4\}$.
1. **Show that all four roots are real and negative:**

   We start with the polynomial:
   \[
   f(x) = x^4 + 14x^3 + 52x^2 + 56x + 16
   \]
   We can factorize it as follows:
   \[
   x^4 + 14x^3 + 52x^2 + 56x + 16 = (x^2 + 7x + 4)^2 - 5x^2
   \]
   This can be rewritten as:
   \[
   (x^2 + (7 - \sqrt{5})x + 4)(x^2 + (7 + \sqrt{5})x + 4) = 0
   \]
   Solving for the roots, we get:
   \[
   x = \left\{\frac{\sqrt{5} - 7 \pm \sqrt{38 - 14\sqrt{5}}}{2}, \frac{-\sqrt{5} - 7 \pm \sqrt{38 + 14\sqrt{5}}}{2}\right\}
   \]

   To show that \(\frac{\sqrt{5} - 7 \pm \sqrt{38 - 14\sqrt{5}}}{2} < 0\), we need to prove:
   \[
   7 - \sqrt{5} > \pm \sqrt{38 - 14\sqrt{5}}
   \]
   Squaring both sides, we get:
   \[
   (7 - \sqrt{5})^2 > 38 - 14\sqrt{5}
   \]
   Simplifying, we have:
   \[
   49 - 14\sqrt{5} + 5 > 38 - 14\sqrt{5}
   \]
   Which simplifies to:
   \[
   54 > 38
   \]
   This is obviously true.

   Similarly, to show that \(\frac{-\sqrt{5} - 7 \pm \sqrt{38 + 14\sqrt{5}}}{2} < 0\), we need to prove:
   \[
   \sqrt{5} + 7 > \pm \sqrt{38 + 14\sqrt{5}}
   \]
   Squaring both sides, we get:
   \[
   (\sqrt{5} + 7)^2 > 38 + 14\sqrt{5}
   \]
   Simplifying, we have:
   \[
   5 + 14\sqrt{5} + 49 > 38 + 14\sqrt{5}
   \]
   Which simplifies to:
   \[
   54 > 38
   \]
   This is also obviously true.

   Thus, all four roots are real and negative.

2. **Find the smallest possible value of \(|z_az_b + z_cz_d|\):**

   We note that:
   \[
   \frac{x^4}{16} = \left(\frac{14x^3}{56x}\right)^2
   \]
   This motivates us to find \(f(2x)\). We find that:
   \[
   \frac{1}{16}f(2x) = x^4 + 7x^3 + 13x^2 + 7x + 1
   \]
   Because the coefficients are symmetric and the quartic is monic, this implies that:
   \[
   2x = \frac{16}{2x} \implies x = \frac{4}{x}
   \]
   Thus, the roots come in pairs that multiply to \(4\). Let:
   \[
   z_1 = -r_1, z_2 = -r_2, z_3 = -r_3, z_4 = -r_4
   \]
   Because \(z_1, z_2, z_3, z_4\) are negative, \(r_1, r_2, r_3, r_4\) are positive. Now WLOG:
   \[
   r_3 = \frac{4}{r_1}, r_4 = \frac{4}{r_2}
   \]
   Notice that the possible values of \(|z_az_b + z_cz_d| = |(-r_a)(-r_b) + (-r_c)(-r_d)| = r_a r_b + r_c r_d\) are:
   \[
   r_1 r_2 + r_3 r_4 = r_1 r_2 + \frac{16}{r_1 r_2}, \quad r_1 r_3 + r_2 r_4 = 4 + 4 = 8, \quad r_1 r_4 + r_2 r_3 = 4\left(\frac{r_1}{r_2} + \frac{r_2}{r_1}\right)
   \]
   We have that:
   \[
   r_1 r_2 + \frac{16}{r_1 r_2} \geq 8 \quad \text{(by AM-GM)}
   \]
   and:
   \[
   4\left(\frac{r_1}{r_2} + \frac{r_2}{r_1}\right) \geq 8 \quad \text{(by AM-GM)}
   \]
   Thus, the smallest possible value of \(|z_az_b + z_cz_d|\) is:
   \[
   \boxed{8}
   \]