Match Equivalent Expressions: (x+6)(x+8) and Related Forms

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

$(\textcolor{red}{a}+\textcolor{blue}{b})(c+d)=\textcolor{red}{a}c+\textcolor{red}{a}d+\textcolor{blue}{b}c+\textcolor{blue}{b}d$

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. $(x+6)(x+8) \\ x\cdot x+x\cdot 8+6\cdot x+6\cdot8\\ x^2+8x+6x+48\\ \boxed{x^2+14x+48}\\$

2. $(6+x)(8-x) \\ \downarrow\\ (6+x)\big(8+(-x)\big) \\ 6\cdot 8+6\cdot (-x)+x\cdot 8+x\cdot(-x)\\ 48-6x+8x-x^2\\ \boxed{48+2x-x^2}\\$

3. $(x+x)(6+8) \\ 2x\cdot14\\ \boxed{28x}$

   In the last expression we simplified above, we first combined like terms in each of the expressions within parentheses, therefore in this case there was no need to use the extended distribution law mentioned at the beginning of the solution to simplify the expression.

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

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

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

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

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