The tiles represent quantities thus:
The x^2 tile is a large square; the x tile is a narrow rectangle the same length as the side of the x^2 tile; the unit tile representing the number 1 is a small square which has a side length equal to the width of the x tile.
The tiles are thin and are coloured red on one side for negative quantities and, for the purpose of illustration, are green when flipped over to represent positive quantities. When two tiles of exactly the same shape and size but with opposite colours are placed together, they form a zero pair and the pair can be removed.
If we take the polynomial x^2+4x-5 we can use the tiles to factorise the expression.
We now need to position the tiles into a rectangle so that the sides of the rectangle represent the correct factors of the polynomial (quadratic). We have one large green square x^2 tile and we need to pack 4 green x tiles and 5 red unit tiles on two sides of the square to form a rectangular shape.
Let's put the x^2 tile in the top left corner of the rectangle. We then have to align the x tiles along the right side of the x^2 tile and along the bottom of the tile. So, we could take 2 x tiles and put then on the right and 2 along the bottom. That would allow us to put 4 unit tiles to fill in the bottom right corner of the rectangle. Unfortunately we have 5 unit tiles, and nowhere to place the 5th tile.
Also, if we take 3 x tiles and line them up to the right side of the square, and align one x tile under the square we can only place 3 unit tiles in the bottom right corner. That's even worse!
What to do?
Remember that if we use two tiles of the same size and shape but opposite colours, they pair as zero. So instead of using only 4 x tiles we can use 6, where the extra tiles form a zero pair. Now we can do the geometry. We align 5 x tiles on the right side of the x^2 tile and we align one x tile along the bottom. The unit tiles now fit under the 5 x tiles on the right and the rectangle is complete!
But we know one of the additional x tiles has to be red side up and the other green side up. Remember the sides of the rectangle are the factors, so the tile added to the four on the right must be the same colour, green. The one along the bottom must be red.
The result is that we have an x tile (the top side of the big square) and 5 green unit tiles because the x tiles are green. That represents x+5. On the left side we have one x tile and a red unit square because the x tile at the bottom is red. This represents x-1. So the factors are (x+5) and (x-1).
By reversing all the colours we see that -(x+5) and -(x-1) are also factors.
