Roots of a Quartic

Algebra Level 3

$\large f(x)^2+3f(x)+2$

If $f(x)= x^2-8x$ , then the equation above has 4 distinct real roots of the form $4 \pm{\sqrt{a}}$ for real value $a$ .

Denote the sum of all values of $a$ as $S$ . Evaluate $\dfrac {1740}S$ .