roots of $f(z)=z^4+8z^3+3z^2+8z+3=0$ in the right half plane This is a question in Ahlfors in the section on the argument principle: How many roots of the equation $f(z)=z^4+8z^3+3z^2+8z+3=0$ lie in the right half plane?
He gives a hint that we should "sketch the image of the imaginary axis and apply the argument principle to a large half disk."
Since $f$ is an entire function, I think I understand that the argument principle tells us that for any closed curve $\gamma$ in $\mathbb{C}$, the winding number of $f(\gamma)$ around 0 is equal to the number of zeros of $f$ contained inside $\gamma$.
How would you go about actually applying the hint though? I am having trouble figuring out what the image of a large half disk under $f$ would look like.

The hint was to sketch the image of the imaginary axis. You can parametrize the imaginary axis as $\gamma(t) = it$. $f(it) = t^4-3t^2+3 + (-8t^3 + 8t)i.$ Do you recall how to plot the parametrized curve $(t^4-3t^2+3,-8t^3+8t)$ from calculus? 
