${(x+a)}{(x+b)}$ $\,=\,$ $x^2+(a+b)x+ab$
The expansion of special product of binomials $x+a$ and $x+b$ can be derived in algebraic method by the multiplication of algebraic expressions.
Multiply each term of first binomial with the second binomial to perform multiplication of algebraic expressions.
${(x+a)}{(x+b)}$ $\,=\,$ $x \times (x+b)$ $+$ $a \times (x+b)$
Now, multiply each term of the second binomial by its multiplying factor.
$=\,$ $x \times x$ $+$ $x \times b$ $+$ $a \times x$ $+$ $a \times b$
Now, write product of the terms in an order to obtain the special product of binomials $x+a$ and $x+b$.
$=\,$ $x^2$ $+$ $xb$ $+$ $ax$ $+$ $ab$
$=\,$ $x^2$ $+$ $bx$ $+$ $ax$ $+$ $ab$
$=\,$ $x^2$ $+$ $ax$ $+$ $bx$ $+$ $ab$
$x$ is a common multiplying factor in two terms of the expression. So, take it common from them to express the expansion of $(x+a)(x+b)$ formula in algebraically.
$\,\,\, \therefore \,\,\,\,\,\,$ ${(x+a)}{(x+b)}$ $\,=\,$ $x^2$ $+$ $(a+b)x$ $+$ $ab$
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