Notes on the Proof of Theorem 4.8

$$\begin{array}{rl}T(x)-2T\left(\frac{x}{2}\right)& =x\log x+O(x)-2(\frac{x}{2}\log \frac{x}{2}+O(x))\\ & =x\log x-x\log \frac{x}{2}+O(x)\\ & =x\log x-x(\log x-\log 2)+O(x)\\ & =x\log x-x\log x+x\log 2+O(x)\\ & =x\log 2+O(x)\\ & =O(x).\end{array}$$