$\overrightarrow{a}=\hat{i}+\hat{j}+2\hat{k}$

$\overrightarrow{b}=-\hat{i}+2\hat{j}+3\hat{k}$

$\overrightarrow{a}+\overrightarrow{b}=3\hat{j}+5\hat{k};\overrightarrow{a}·\overrightarrow{b}=-1+2+6=7$

Now $\left(\left(\overrightarrow{a}\times \left(\left(\overrightarrow{a}-\overrightarrow{b}\right)\times \overrightarrow{b}\right)\right)\times \overrightarrow{b}\right)$

$=\left(\left(\overrightarrow{a}\times \left(\overrightarrow{a}\times \overrightarrow{b}-\overrightarrow{b}\times \overrightarrow{b}\right)\right)\times \overrightarrow{b}\right)$

$=\left(\overrightarrow{a}\times \left(\overrightarrow{a}\times \overrightarrow{b}-0\right)\right)\times \overrightarrow{b}$

$=\left(\overrightarrow{a}\times \left(\overrightarrow{a}\times \overrightarrow{b}\right)\right)\times \overrightarrow{b}$

$=\left(\left(\overrightarrow{a}·\overrightarrow{b}\right)\overrightarrow{a}-\left(\overrightarrow{a}·\overrightarrow{a}\right)\overrightarrow{b}\right)\times \overrightarrow{b}$

$=\left(\overrightarrow{a}·\overrightarrow{b}\right)\left(\overrightarrow{a}\times \overrightarrow{b}\right)-\left(\overrightarrow{a}·\overrightarrow{a}\right)\left(\overrightarrow{b}\times \overrightarrow{b}\right)$

$=\left(\overrightarrow{a}·\overrightarrow{b}\right)\left(\overrightarrow{a}\times \overrightarrow{b}\right)$

Here, $\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& 2\\ -1& 2& 3\end{array}\right|=-\hat{i}-5\hat{j}+3\hat{k}$

$\therefore \left(\overrightarrow{a}·\overrightarrow{b}\right)\left(\overrightarrow{a}\times \overrightarrow{b}\right)=7\left(-\hat{i}-5\hat{j}+3\hat{k}\right)$

So, $\left(\overrightarrow{a}+\overrightarrow{b}\right)\times \left(7\left(-\hat{i}-5\hat{j}+3\hat{k}\right)\right)$

$7\left(0\hat{i}+3\hat{j}+5\hat{k}\right)\times \left(-\hat{i}-5\hat{j}+3\hat{k}\right)$

$=7\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 0& 3& 5\\ -1& -5& 3\end{array}\right|$

$=7\left[34\hat{i}-\left(5\right)\hat{j}+\left(3\hat{k}\right)\right]$

$=7\left[34\hat{i}-5\hat{j}+3\hat{k}\right]$