Match Equivalent Expressions: (a+b)(c+d) and Its Expanded Forms

Let's simplify the given expressions, open the parentheses using the extended distribution law:

$(\textcolor{red}{x}+\textcolor{blue}{y})(m+r)=\textcolor{red}{x}m+\textcolor{red}{x}r+\textcolor{blue}{y}m+\textcolor{blue}{y}r$

In the formula template for the above distribution law, we take by default that the operation between the terms inside of the parentheses is addition. Note that the sign preceding the term is an inseparable part of it. Furthermore we will apply the laws of sign multiplication to our expression. We will then open the parentheses using the above formula, where there is an addition operation between all terms.

Proceed to simplify each of the expressions in the given problem, whilst making sure to open the parentheses using the mentioned distribution law, the commutative law of addition and multiplication and combining like terms (if there are like terms in the expression obtained after opening the parentheses):

1. $(a+b)(c+d) \\ \boxed{ac+ad+bc+bd}\\$

2. $(a+c)(b+d) \\ \boxed{ab+ad+cb+cd}$

3. $(a+d)(c+b) \\ \boxed{ac+ab+dc+db}$

   In all expressions where we performed a multiplication operation between the expressions in the above parentheses, the result of the multiplication (obtained after applying the mentioned distribution law) yielded an expression where the terms cannot be combined. This is due to the fact that all terms in the resulting expression are different from each other ( All variables in like terms need to be identical and have the same exponent)

4. After applying the commutative law of addition and multiplication we observe that:

   The simplified expression in 1 matches the expression in option A,

   The simplified expression in 2 matches the expression in option C,

   The simplified expression in 3 matches the expression in option B,

Therefore the correct answer (among the suggested options) is answer C.