The whole is the sum of its parts. This is a property that we learn and rely heavily upon in our study of Geometry, but we practice its usefulness in Algebra, especially when learning about polynomials. Most students of algebra can look at the figure on the left above and calculate the area of the blue shaded region. How do they do it? You may ask. Or maybe you wonder why I even ask this question, maybe it is quite obvious to you. However there is another way, a better way even, to compute the area shaded in blue. This way paves the path to understanding some simple polynomial patterns.
First, let's look at the left figure. Most algebra students will practice the age old property mentioned above and "break" the blue section into two rectangles. If I can find the area of each rectangle, then the sum of those two rectangles is the area of the blue region. Sure enough, my friend. That is true. What if instead of breaking the blue region into two smaller rectangles, I chose to observe the region as the displacement between the large square and the small square? That is to say, the blue region is the difference of 5 squared and 3 squared. Do you see it? Another way to view this shape.
The two procedures that result from observing the same space as two different expressions grants us the most wonderful equation: 5 squared - 3 squared = 5 (5 - 3) + (5-3)(5-3). Why do I insist on this equation being mind blowing? Apply this thinking and procedure to the squares on the right: the ones with no numbers, but instead letters representing these distances. What do you discover?
If you took the time to study this you discovered not just the simple pattern that holds true for the factors of all difference of squares, but also how we can rationalize denominators quickly, and calculate the product of a complex number and its conjugate simply. A whole new world of calculation and reason is opened to you. You just need to be willing to view the same shape in multiple ways.