Root Locus and Imaginary Axis
I don't know why "k" has been introduced halfway through your algebra but, ignoring it until told otherwise, you are nearly there. Your final formula reduces to a real term in the denominator (because that is what multiplying top and bottom by the complex conjugate does) so just concentrate on the numerator. 

Expand that out and then quite simply equate the sum of all the real terms in the numerator to zero. Ignore everything else because all the real terms equating to zero marks the position on the root locus that it crosses the imaginery axis.

Then drill down to find \$\omega\$. And a happy xmas to you.