(a)[4]
The equation $z^3 + 2z + a = 0$, where $a$ is real, has root $-1 + (\sqrt{5})i$. Show your working to find $a$, and then write down the other complex root of this equation.
(b)[4]
Given that the complex number $w$ has modulus $1$ and argument $2\theta$ radians, show that $\frac{w - 1}{w + 1} = i\tan \theta$.