Using algebra prove that the square of any odd number is always 1 more than a multiple of 8

Start off with (x+1) (x-1). If x is 5 for example, use 4 and 6. For any odd number x, there are two even numbers next to it on the number line. One of these will be at least a multiple of 2 and one at least a multiple of 4. If you go to a number line, you see that in every group of 4 natural numbers has 2 odds and 2 evens. These evens include that multiple of 4 (or a greater power of 2) and that multiple of 2 (or greater power of 2).