Given: $\overrightarrow a\times (2\hat i+3\hat j+4\hat k)=(2\hat i+3\hat j+4\hat k)\times \overrightarrow b$
and $|\overrightarrow a+\overrightarrow b|=\sqrt {29}$
$\Rightarrow\:(\overrightarrow a+\overrightarrow b)\times (2\hat i+3\hat j+4\hat k)=0$
$\Rightarrow\:(\overrightarrow a+\overrightarrow b) $ is parallel to $(2\hat i+3\hat j+4\hat k)$
and hence
$\overrightarrow a+\overrightarrow b=\large\frac{2\hat i+3\hat j+4\hat k}{\sqrt {29}}.$$\sqrt {29}=2\hat i+3\hat j+4\hat k$
Now
$(\overrightarrow a+\overrightarrow b).(-7\hat i+2\hat j+3\hat k)$
$=-14+6+12=4$