jmc

It's certainly possible to just calculate the complex number ${\omega }^2+2\omega -8$ by just plugging in the value of $\omega$, but it's computationally simpler to use the fact that $|ab|=|a|\cdot |b|$ and our knowledge of factoring quadratics: $$\begin{array}{rl}|{\omega }^2+2\omega -8|& =|(\omega -2)(\omega +4)|\\ & =|\omega -2|\cdot |\omega +4|\\ & =|-3+4i|\cdot |3+4i|\\ & =\sqrt{(-3{)}^2+4^2}\sqrt{3^2+4^2}\\ & =\boxed{25}\end{array}$$