how we find the root of
x^2+4x+1...
where x=exp(jw);

Use the quadratic equation?

x = \frac{-4 \pm \sqrt{4^2 - 4}}{2}

x =-2 \pm \sqrt{3}

And then, you substitute back x = exp(jw) though I'm not sure what you mean by that... is it:

x = e^{iw}

?