How do I fix slow graphics on this old Intel 82Q35 Express powered machine?

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http://askubuntu.com – My HP dc7800 is very slow. It has 4GB memory and is running Ubuntu 12.04 64 bits. It also shows artefacts when scrolling web pages. This problem is confirmed Googling for its chipset, the Intel Corporation 82Q35 Express. However, the posts marked [Solved], suggested to get rid of the machine. I like the dc7800 because of the ultra small form factor, so I am asking if dumping the HP is really the last resort. (HowTos)