Proving time-complexity analysis for merge-sort-like algorihtm
There is no reason to calculate a closed form solution to this recurrence. Indeed, this is not even possible, due to the following reason:


$O(n)$ is not a closed form function. It just signifies that the recurrence holds with $O(n)$ replaced by some function $f(n)$ which belongs to the class $O(n)$.
It is not clear what happens when you plug in $n$ which is not divisible by 4.
A related issue is that there are no initial conditions - the recurrence seemingly never bottoms out.


Any reasonable way to address the latter two concerns will lead to a recurrence which is amenable to the Akra&ndash;Bazzi method. According to this method, we first need to calculate the value of $p$ satisfying
$$ (1/4)^p + (3/4)^p = 1. $$
The solution to this equation is $p = 1$. The asymptotic solution is then
$$
T(n) = O\left(n\left(1 + \int_1^n \frac{O(u)}{u^2} \, du \right)\right) = O(n\log n).
$$