It is possible to use the distributive property to simplify the expression?

If so, what is its simplest form?

$(a+c)(4+c)$

We simplify the given expression by** opening the parentheses using the **__extended distributive property__:

$(\textcolor{red}{x}+\textcolor{blue}{y})(t+d)=\textcolor{red}{x}t+\textcolor{red}{x}d+\textcolor{blue}{y}t+\textcolor{blue}{y}d$Keep in mind that in the distributive property formula mentioned above, **we assume that the operation between the terms inside the parentheses is an addition operation**, therefore, of course, we will not forget that __the sign of the term's coefficient is ery important__.

We will also apply the rules of multiplication of signs, so we can present any expression within parentheses that's opened with the distributive property as an expression with addition between all the terms.

In this expression we only have addition signs in parentheses, therefore we go directly to opening the parentheses,

**We start **by opening the parentheses:

$(\textcolor{red}{x}+\textcolor{blue}{c})(4+c)\\
\textcolor{red}{x}\cdot 4+\textcolor{red}{x}\cdot c+\textcolor{blue}{c}\cdot 4+\textcolor{blue}{c} \cdot c\\
4x+xc+4c+c^2$To simplify this expression, we use the power law for multiplication between terms with identical bases:

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

In the next step __like terms come into play__.

We define like terms as __terms in which the variables (in this case, x and c) have identical powers__ (in the absence of one of the variables from the expression, we will refer to its power as zero power, this is because __raising any number to the power of zero results in 1__).

We will also use the substitution property, and we will order the expression from the highest to the lowest power from left to right (we will refer to the regular integer as the power of zero),

Keep in mind that in this new expression there are four different terms, this is because there is not even one pair of __terms__ in which the variables (different) have the same power. Also it is already ordered by power, __therefore the expression we have is the final and most simplified expression:__$\textcolor{purple}{4x}\textcolor{green}{+xc}\textcolor{black}{+4c}\textcolor{orange}{+c^2 }\\
\textcolor{orange}{c^2 }\textcolor{green}{+xc}\textcolor{purple}{+4x}\textcolor{black}{+4c}\\$**We highlight the **__different__ terms using colors and, as emphasized before, we make sure that the main sign of the term is correct.

__We use the substitution property for multiplication to note that the correct answer is option A.__

Yes, the meaning is $4x+cx+4c+c^2$