For convenience I've attached both the question and my attempt at the solution. Please see figures attached.

From the quadratic equation for the desired phase margin, I selected the w = 2.682 solution because I thought that we couldn't have negative values for w. (i.e. not physical)

However the correct solution comes about when w = +1.491, but the solution to my equation indicates w = -1.491.

How do we know when a negative frequency indicates an invalid solution, or a valid one where we can simply neglect the negative sign?

 

I believe it's a simple sign error in your equations:

You have

\(-130^o=-90^o-atan(\frac{\omega}{4-\omega^2})\)

which then becomes

\(-40^o=-atan(\frac{\omega}{4-\omega^2})\)

then

\(40^o=atan(\frac{\omega}{4-\omega^2})\)

then

\(tan(40^o)=(\frac{\omega}{4-\omega^2})\)

and so on ....