A difficulty in understanding a step in a solution.

Question

Here is the solution:

But I could not understand how the last term in the fourth line came from the line before it, could anyone explain this for me please?

EDIT: I have highlighted the problem:

Answer 1

\begin{align*} \operatorname{ln}n\frac{4n^2+5n}{4n^2+8n+4} &= \operatorname{ln}n\frac{4n^2+5n}{4n^2+8n+4} - \operatorname{ln}n + \operatorname{ln} n \\ &= \operatorname{ln}n\left(\frac{4n^2+5n}{4n^2+8n+4} - 1\right) + \operatorname{ln}n \end{align*}

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Edit: For the new edited question note that

$$n\operatorname{ln}n\frac{4n+5}{4n^2+8n+4} = \operatorname{ln}n\frac{4n^2+5n}{4n^2+8n+4}$$

This is the same situation as $$xy\frac{a}{b} = y\frac{xa}{b}$$

Answer 2

It's the same as in $xy = x(y-1)+x$. Here $x=\ln n$ and $y$ is the fraction.