Expand (7x+4)(3x+4): Step-by-Step Binomial Multiplication

Solve the following problem:

$(7x+4)(3x+4)=$

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 automatically assume that the operation between the terms inside of the parentheses is addition. Furthermore remember that the sign preceding the term is an inseparable part of it. By applying the rules of sign multiplication we can present any expression inside of the parentheses. We will open the parentheses using the above formula, first as an expression where an addition operation exists between all terms. In this expression given that all terms are positive we'll proceed directly to opening the parentheses,

$$Let's open the parentheses:

$(\textcolor{red}{7x}+\textcolor{blue}{4})(3x+4)\\ \textcolor{red}{7x}\cdot3x+ \textcolor{red}{7x}\cdot4+\textcolor{blue}{4}\cdot 3x +\textcolor{blue}{4}\cdot4\\ 21x^2+28x+12x+16$

In calculating the above multiplications, we used the multiplication table and 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 will define as terms where the variable (or variables each separately) In this case x, have identical exponents (in the absence of one of the variables from the expression, we'll treat its exponent as zero power since raising any number to the zero power yields the result 1) We'll apply the commutative property of addition, furthermore we'll arrange (if needed) the expression from highest to lowest power from left to right (treating the free number as zero power):
$\textcolor{purple}{21x^2}\textcolor{green}{+28x}\textcolor{green}{+12x}+16\\ \textcolor{purple}{21x^2}\textcolor{green}{+40x}+16$

In the combining of like terms performed above, 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,

Therefore the correct answer is answer B.