Step-by-Step Solution
Key Concept: Use substitution to reduce a quartic equation in $z$ to a quadratic equation, then solve by factoring
Given $z^4 + 4z^2 + 3 = 0$, substitute $w = z^2$ to get $w^2 + 4w + 3 = 0$, which factors as $(w+1)(w+3) = 0$. Thus $w = -1$ or $w = -3$. When $z^2 = -1$, we get $z = \pm i$. When $z^2 = -3$, we get $z = \pm i\sqrt{3}$. The sum of all roots is $|i| + |-i| + |i\sqrt{3}| + |-i\sqrt{3}| = 1 + 1 + \sqrt{3} + \sqrt{3} = 2 + 2\sqrt{3}$.
Correct Answer: 2, 3 + 2iā3