${(x+a)}{(x+b)}$ $\,=\,$ $x^2+(a+b)x+ab$
The $(x+a)(x+b)$ algebraic identity can be derived geometrically by the concept of areas of rectangle and square.
The geometrical approach splits a rectangle as a square and three different small rectangles. Now, calculate area of every geometric shape mathematically.
It is derived that the area of a whole rectangle is ${(x+a)}{(x+b)}$. Actually, the same rectangle is divided as a square and three small different rectangles. So, the area of rectangle should be equal to the sum of the areas of one square and three different rectangles.
${(x+a)}{(x+b)}$ $\,=\,$ $x^2+bx+xa+ba$
$\implies$ ${(x+a)}{(x+b)}$ $\,=\,$ $x^2+bx+ax+ab$
$\implies$ ${(x+a)}{(x+b)}$ $\,=\,$ $x^2+ax+bx+ab$
$\,\,\, \therefore \,\,\,\,\,\,$ ${(x+a)}{(x+b)}$ $\,=\,$ $x^2+(a+b)x+ab$
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