Before root the Toshiba Satellite S875-S7140，you must know the following
1. Not every phone can be rooted，then you must make sure other people can do it too.
2. Backup the phone, as unlocking the bootloader will reset everything.
3. Rooting voids your warranty.
4. In my experience that isn’t foolproof.
1, Go to http://www.htcdev.com/ and follow the instructions to get into bootloader mode. You will disconnect your Toshiba Satellite S875-S7140 from PC, turn it off, press and hold Volume Down and the Power button to access the Bootloader.
2, Use the volume keys to select fastboot mode, and select it with power. When Toshiba Satellite S875-S7140 does its thing, connect it to the PC again.
3, On your PC, go to the folder with all the ADB and Fastboot files and open a command prompt. Type fastboot devices and hit enter. The Toshiba Satellite S875-S7140 should show up as a device ID.
4, Enter fastboot oem get_identifier_token. This will spit out a big block of text, which you will then copy and paste into the HTC Dev site when it asks for it. Wait a bit, and HTC will send you the unlock token. Place it in the Fastboot folder.
5, With the Toshiba Satellite S875-S7140 still in Fastboot mode, type fastboot flash unlocktoken Unlock_code.bin. Now you’re unlocked, so it’s time to root. Reboot the phone normally and download the SuperUser zip file from Koush. Transfer it to Toshiba Satellite S875-S7140.
6, Download TWRP recovery for the Toshiba Satellite S875-S7140 and place it in the directory with your Fastboot and ADB files.
7, Open another command prompt and get your device back into Fastboot mode just like above. Type fastboot flash recovery twrp.img and hit enter.
8, Turn off the Toshiba Satellite S875-S7140 and turn it on in Bootloader mode by holding the Volume Down + Power button. Select Recovery and wait for Toshiba Satellite S875-S7140 to load.
9, In Recovery, go to install and select the SuperUser zip. The zip will flash, and that’s it. Then your Toshiba Satellite S875-S7140 rooted.
10, If the above can not solve the problem then google for “root Toshiba Satellite S875-S7140” and you’ll find what you’re looking for.