Multiplying builds a rectangle out of pieces. Factoring runs the film backwards — you are handed the pieces and asked to rebuild the rectangle. Two moves do almost all the work: pull out the greatest common factor (find the shared height), and, when four terms have nothing in common, factor by grouping (find the shared width). Step through each example one line at a time; the picture on the left shows what the algebra is actually doing.
The picture
Blocks are drawn schematically — they show which factor is shared, not exact sizes.
Factoring, one line at a timepick an example below