jmc

The greatest common factor of the two coefficients is $11$, and the greatest power of $z$ that divides both terms is $z^{17}$. So, we factor $11z^{17}$ out of both terms:

$$\begin{array}{rl}55z^{17}+121z^{34}& =11z^{17}\cdot 5+11z^{17}\cdot 11z^{17}\\ & =\boxed{11z^{17}(5+11z^{17})}\end{array}$$