jmc

It's certainly possible to just calculate the complex number $2{\omega }^2-4\omega -30$ by just plugging in the value of $\omega$, but it's computationally simpler to use the fact that $|ab|=|a||b|$ and our knowledge of factoring quadratics: $$\begin{array}{rl}|2{\omega }^2-4\omega -30|& =|2(\omega -5)(\omega +3)|\\ & =2|\omega -5|\cdot |\omega +3|\\ & =2|-4-5i|\cdot |4-5i|\\ & =2\sqrt{(-4{)}^2+5^2}\sqrt{4^2+5^2}\\ & =\boxed{82}\end{array}$$