The full electron configuration of silicon is 1s22s22p63s23p2 To find its noble gas shorthand, look at the full electron configuration and see how many filled shells there are. For silicon, all but the outermost shell is filled. So the number of filled shells is 2.
Now look at the periodic table and find the noble gas at the end of the period the same as the number of filled shells. The noble gas at the end of Period 2 is neon, Ne.
If you replace all of the electron configuration for the filled shells with the noble gas's symbol, which is in square brackets, you get to the answer at the top of the page.
Note: When you get to the transition elements, you will have unfilled d-sub shell electrons to take into account. These need showing explicitly.