Sorry I should have read your initial post more carefully.  I see you are running ubuntu 10.4, and I have ubuntu 10.10, so there may be some important differences here.

From the desktop I run System->Administration->Synaptic Package Manager,
from this application I enter "qt4" in the Quick Search box.  In this window I can now select "qt4-demos" among many other options.  But I do not see qt-x11-opensource-src-4.4.3 among the available packages for my ubuntu machine.  And the qt version is 4.7.0 on my machine, not 4.4.3 as seen on yours.

Can you run the Synaptic Package Manager on you ubuntu machine?  What do you see if you type "qt demos" in the Quick Search box?  What is the version of Qt mentioned there?  If the version of Qt available for your ubuntu 10.4 machine is too old, you should take Hanchuan Peng's advice and build the latest Qt from source.