\(\begin{array}{*{20}{l}}
{\left( {2 + i} \right)z + \frac{{2\left( {1 + 2i} \right)}}{{1 + i}} = 7 + 8i}\\
\begin{array}{l}
 \Leftrightarrow z\left( {2 + i} \right)\left( {1 + i} \right)\left( {1 - i} \right)\\
 + 2\left( {1 + 2i} \right)\left( {1 - i} \right) = \left( {7 + 8i} \right)\left( {1 + i} \right)\left( {1 - i} \right)
\end{array}\\
{ \Leftrightarrow \left( {4 + 2i} \right)z + 2 - 2i + 4i + 4 = 14 + 16i}\\
{ \Leftrightarrow \left( {4 + 2i} \right)z = 8 + 14i}\\
{ \Leftrightarrow z = \frac{{8 + 14i}}{{4 + 2i}} = 3 + 2i}\\
{ \Rightarrow \left| {\rm{w}} \right| = \left| {z + i + 1} \right| = \left| {4 + 3i} \right| = 5}
\end{array}\)