Match Binomial Products: (2x+y)(x+2y) and Related Expressions

Let's simplify the given expressions, open the parentheses using the expanded 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, so we won't forget of course that the sign preceding the term is an inseparable part of it, and we'll also apply the rules of sign multiplication and thus we can present any expression in parentheses, which we'll open using the above formula, first, as an expression where addition operation exists between all terms (if needed),

We will therefore simplify each of the expressions in the given problem, while being careful about the above, first opening the parentheses using the mentioned 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. $(2x+y)(x+2y) \\ 2x\cdot x+2x\cdot 2y+y\cdot x+y\cdot2y\\ 2x^2+4xy+xy+2y^2\\ \boxed{2x^2+5xy+2y^2}$

2. $(2x+2y)(x+y) \\ 2x\cdot x+2x\cdot y+2y\cdot x+2y\cdot y\\ 2x^2+2xy+2yx+2y^2\\ \boxed{2x^2+4xy+2y^2}$

3. $(2x-y)(x-2y) \\ \downarrow\\ \big(2x+(-y)\big)\big(x+(-2y)\big) \\ 2x\cdot x+2x\cdot(-2y)+(-y)\cdot x+(-y)\cdot(-2y)\\ 2x^2-4xy-yx+2y^2\\ \boxed{2x^2-5xy+2y^2}$

   Now, we'll use the commutative law of addition and multiplication to notice 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 A.