Determine the number of zeros for $4z^3-12z^2+2z+10$ in $\frac{1}{2}<|z-1|<2$. I'm faced with the problem in the title

Determine the number of zeros for $4z^3-12z^2+2z+10$ in the annulus $\frac{1}{2}<|z-1|<2$. 

Clearly this requires a nifty application of Rouche's Theorem. Why this isn't  so easy for me is because the annulus isn't centered at the origin. In those cases in which it is centered at the origin, it's a simple plugging in of numbers. Could anybody suggest a next step to take?

Well I just figured out what to do while writing this, so I'm going to write the trick out here for anybody that might run across the same problem I had.
Just write the polynomial $4z^3-12z^2+2z+10$ as a Taylor Polynomial centered at $z=1$. You don't actually need to use Taylor's theorem for this - using algebra could do it also.
Doing that, you get $4(z-1)^3-10(z-1)+4$. With a simple substitution $s=z-1$, we get $4s^3-10s+4$, and from there it's trivial.
