In the diagram above we have a large square whose sides have length 26 inches and we want to find its area.
The usual way you might do this is to multiply 26 by 26 and get the answer:

The area is square inches.

But this means that you have to multiply a rather unpleasant number (26) by itself. There is another way you might proceed.
Notice (in the diagram at left) that the black square in the lower left has side 20 inches and so its area is 400 square inches.
The two blue rectangles each have sides of length 20 inches and 6 inches and so they have area 120 square inches.
The red square has side 6 inches and so it has area 36 square inches. Thus the are of the large square is:

400+120+120+36 = square inches.

A third way is illustrated by the diagram at right. First, notice that the complete square at right has sides of total length 30 inches
so it has area 900 square inches. Now look at the big top rectangle (including the blue and red rectangles).
It has dimensions 30 inches by 4 inches. Similarly the big right rectangle has dimensions 4 inches by 30 inches.
Now suppose I remove the top rectangle. Then I add the red rectangle back.
This leaves me the black square (which has sides of length 26 inches) plus a copy of the big right rectangle.
Now, if I remove the big right rectangle I am left with the square with sides of 26 inches.

Thus the area of the black square is 900-120+16-120= square inches

You can see which approach you prefer by choosing one of the methods described above to do these problems:

$31^2=$ .

$19^2=$ .

$27^2=$ .

$105^2=$ .

$303^2=$ .

$82^2=$ .

If you look carefully at what we have done you will see that the method illustrated by the diagram at left shows that

$(x+y)^2=x^2+2xy+y^2$

while the method illustrated by the diagram at right shows that

$(x-y)^2=x^2-2xy+y^2$.