Expand the Binomial Product: (2x+3y)(2z+12m) Step by Step

Solve the following problem:

$(2x+3y)(2z+12m)=$

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

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

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. Remember that the sign preceding the term is an inseparable part of it. By applying the laws of sign multiplication we can present any expression inside of the parentheses. We open the parentheses using the above formula, first as an expression where an addition operation exists between all terms. In this expression it's clear that all terms have a plus sign prefix. Therefore we directly proceed to opening the parentheses,

$$Start by opening the parentheses:

$(\textcolor{red}{2x}+\textcolor{blue}{3y})(2z+12m)\\ \textcolor{red}{2x}\cdot 2z+\textcolor{red}{2x}\cdot12m+\textcolor{blue}{3y}\cdot 2z+\textcolor{blue}{3y} \cdot12m\\ 4xz+24xm+6yz+36ym$

In the next step we'll combine like terms, which we define as terms where the variable (or variables, each separately), in this case z,m x and y, have identical exponents. (In the absence of one of the variables from the expression, we'll consider its exponent as zero power, this is due to the fact that any number raised to the power of zero equals 1),

Note that in the expression that we obtained in the last step there are four different terms. This is due to the fact that there isn't even one pair of terms where the (different) variables have the same exponent. Therefore the expression that we already obtained is the final and most simplified expression:
$\textcolor{purple}{ 4xz}\textcolor{green}{+24xm}\textcolor{black}{+6yz}\textcolor{orange}{+36ym }\\$We highlighted the different terms using colors, and as emphasized before, we made sure that the sign preceding the term is an inseparable part of it,

We can conclude that the correct answer is answer D.