A strange equation VI

Algebra Level 3

If one root of the equation \[x^4-4x^3-16x^2-8x+4=0\] can be expressed as \(a+\sqrt{b}+\sqrt{c}+\sqrt{d}\) where \(b\), \(c\) and \(d\) are square-free numbers, then find \(a+bcd\).