Completing the square

Draw x² as a square with sides x. Draw 6x as a strip x long and 6 wide. Cut the strip in half lengthways and put one half on the right of the square and one half along the bottom.

You now have an almost-square with a missing corner. The corner is 3 by 3, so adding 9 completes it: x² + 6x + 9 is a square with sides x + 3.

That is the whole method. Half the strip goes on each side, and the missing corner is that half, squared.