Expand (x+y)(3x+2y): Step-by-Step Binomial Multiplication

Expand the following expression:

$(x+y)(3x+2y)=$

Let's simplify the given expression by opening 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 of the parentheses is addition. We must remember that the sign preceding the term is an inseparable part of it. We will also apply the rules of sign multiplication in order to present any expression in the parentheses, which we will open using the above formula. First as an expression where addition operation exists between all terms. In this expression it's clear that all terms have a plus sign prefix, therefore we'll proceed directly to opening the parentheses,

$$Let's begin to open the parentheses:

$(\textcolor{red}{x}+\textcolor{blue}{y})(3x+2y)\\ \textcolor{red}{x}\cdot 3x+\textcolor{red}{x}\cdot2y+\textcolor{blue}{y}\cdot 3x+\textcolor{blue}{y} \cdot2y\\ 3x^2+2xy+3xy+2y^2$

In calculating the above multiplications, we used the multiplication table as well as the laws of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

In the next step we'll combine like terms, which we'll define as terms where the variable (or variables, each separately), in this case x and y, have identical exponents. (In the absence of one of the variables from the expression, we'll treat its exponent as zero power. This is due to the fact that raising any number to the zero power yields the result 1) We'll apply the commutative law of addition, and proceed to arrange the expression (if needed) from highest to lowest power from left to right (we'll treat the free number as zero power):
$\textcolor{purple}{ 3x^2}\textcolor{green}{+2xy}\textcolor{green}{+3xy}\textcolor{orange}{+2y^2}\\ \textcolor{purple}{ 3x^2}\textcolor{green}{+5xy}\textcolor{orange}{+2y^2}\\$We highlighted the different terms using colors, and as emphasized before, we made sure that the sign preceding the term remains an inseparable part of it,

We therefore concluded that the correct answer is answer A.