Although in the ques it is asked for the value of $n$ for which insertion sort beats merge sort, but solution in the image says for $n$> 43 merge sort beats insertion sort.
So, essentially we need to find $n$ such that following inequality becomes false ( because we want $n$ for which merge sort takes longer) :
$8n^{2} < 64 n\log n$ $\Rightarrow$ $n < 8\log n$
Till n=43, it is
$43 < 8 \ast \log 43$ $\Rightarrow$ $43 < 8 \ast 5.42 = 43.41$
At n=44, it is
$44 < 8 \ast \log 44$ $\Rightarrow$ $44 < 8 \ast 5.45 = 43.67$
For the last case and for all n>43 this inequality fails. Thus, merge sort performs better than insertion sort for n>43 i.e. merge sort beats insertion sort for n>43.
This question just tries to convey the fact that insertion sort is a better choice when our input size is small.
