Match Equivalent Expressions: (12x-5)(y+2) and Related Binomials

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

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$Note that in the formula template for the above distribution law, we take by default that the operation between the terms inside the parentheses is addition, therefore we won't forget of course that the sign preceding the term is an integral part of it, and we will also apply the rules of sign multiplication and thus we can present any expression in parentheses, which we open using the above formula, first, as an expression where addition operation exists between all terms (if required),

We will therefore simplify each of the expressions in the given problem, while being careful about the above, first opening the parentheses using the aforementioned distribution law and then using 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. $(12x-5)(y+2) \\ \downarrow\\ \big(12x+(-5)\big)(y+2)\\ 12x\cdot y+12x\cdot 2+(-5)\cdot y+(-5)\cdot 2\\ \boxed{12xy+24x-5y-10}\\$

2. $(x-12)(5y+2) \\ \downarrow\\ \big(x+(-12)\big)(5y+2)\\ x\cdot 5y+x\cdot 2+(-12)\cdot 5y+(-12)\cdot 2\\ \boxed{5xy+2x-60y-24}\\$

3. $(12x+5)(y-2) \\ \downarrow\\ (12x+5)\big(y+(-2)\big)\\ 12x\cdot y+12x\cdot (-2)+5\cdot y+5\cdot (-2)\\\ \boxed{12xy-24x+5y-10}\\$

   As can be noticed, in all expressions where we started the multiplication between the expressions in parentheses above, the result of multiplication (obtained after applying the aforementioned distribution law) yielded an expression where terms cannot be combined, and this is because all terms in the resulting expression are different from each other (remember that all variables in like terms need to be identical and have the same exponent),

   Now, let's use the commutative law of addition and multiplication to observe that:

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

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

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

Therefore, the correct answer (among the suggested options) is answer B.