Suppose that $T$ is a normal operator on $V$. Show that $\|T(v+w)\|=10$. Suppose $T$ is a normal operator on $V$. Suppose also that $v, w \in V$ satisfy the equations 
$$ \| v \|= \| w \| =2, Tv = 3v, Tw = 4w.$$
Show that $\| T(w+v) \| = 10.$ 
I thought this problem would be easy, but I am stuck.  I began by writing
\begin{align*}
\|T(v+w)\|&=\sqrt{\langle T(v+w),T(v+w)\rangle}\\
&=\sqrt{\langle Tv,Tv\rangle+\langle Tv,Tw\rangle+\langle Tw,Tv\rangle+\langle Tw,Tw\rangle}
\end{align*}
At this point I figure I should use the fact that $T$ is normal (i.e., $TT^*=T^*T$, where $T^*$ is the adjoint of $T$), but I just can't figure out how!  Hints would be appreciated.

First, recall or prove that for a normal operator  eigenvectors of two distinct eigenvalues are orthogonal. 
Second, do as you did,  then replace $Tv$ and $Tw$ by $3v$ and $4w$, respectively, pull out the constants, and simplify.  
