Calculate the Rectangle's Perimeter with Area Expression m²+4m-12

First let's refer to rectangle $ABCD$:

Let's continue and write down the data regarding the rectangle's area and the given side length in mathematical form:

$\begin{cases} \textcolor{red}{A_{ABCD}}= m^2+4m-12 \\ \textcolor{blue}{AB}=m+6\\ \end{cases}$(we'll use colors here for clarity of the solution later)

Now let's remember that the area of a rectangle whose (adjacent) sides are:

$a,\hspace{2pt}b$is:

$A_{\boxed{\hspace{6pt}}}=a\cdot b$

therefore the area of the rectangle in this problem (according to the drawing we established at the beginning of the solution) is:

$A_{ABCD}=AB\cdot AD$

Now we can substitute the previously mentioned data in this area expression to get the equation (for understanding - use the marked colors and the data mentioned earlier accordingly):

$\textcolor{red}{ A_{ABCD}}=\textcolor{blue}{AB}\cdot AD \\ \downarrow\\ \boxed{ \textcolor{red}{ m^2+4m-12}=\textcolor{blue}{(m+6)}\cdot AD}$

Now, let's pause for a moment and ask what is our goal?

Our goal is of course to get the algebraic expression for the rectangle's perimeter (in terms of m), for this we remember that the perimeter of a rectangle is the sum of all its sides' lengths (let's denote the rectangle's perimeter as $P_{ABCD}$),

$$Additionally, remember that opposite sides in a rectangle are equal therefore the given perimeter is:

$P_{ABCD}=AB+AD+BC+DC\\ \begin{cases} AB=DC\\ AD=BC \end{cases}\\ \downarrow\\ \boxed{\textcolor{purple}{P_{ABCD}=2\cdot(AB+AD)} }$

However, we already have the algebraic expression for the rectangle's side $AB$ (from the problem's given data, previously marked in blue), therefore all we need to calculate the rectangle's perimeter is the algebraic expression (in terms of m) for the length of side $AD$,

Let's return then to the equation we reached in the step before addressing the perimeter (highlighted with a square around the equation) and isolate $AD$ from it, we'll do this by dividing both sides of the equation by the algebraic expression that is the coefficient of $AD$, meaning by:$(m+6)$:

$\boxed{ \textcolor{red}{\textcolor{blue}{(m+6)}\cdot AD= m^2+4m-12}} \hspace{4pt}\text{/:}(m+6)\\ \downarrow\\ AD=\frac{m^2+4m-12}{m+6}$

Let's continue and simplify the algebraic fraction we got, we can do this easily by factoring the numerator:

$m^2+4m-12$

We'll use quick trinomial factoring for this (to review quick trinomial factoring rules) and get:

$m^2+4m-12\leftrightarrow\begin{cases} \boxed{?}\cdot\boxed{?}=-12\\ \boxed{?}+\boxed{?}=4\ \end{cases}\\ \downarrow\\ (m+6)(m-2)$

therefore (returning to the expression for $AD$):

$AD=\frac{m^2+4m-12}{m+6} \\ \downarrow\\ AD=\frac{(m+6)(m-2)}{m+6}\\ \downarrow\\ \boxed{AD=m-2}$

In the final stage, after factoring the fraction's numerator, we reduced the fraction,

Let's return then to the expression for the rectangle's perimeter, which we got earlier and substitute in it the algebraic expressions for the rectangle's side lengths that we found, then we'll simplify the resulting expression:

$\boxed{\textcolor{purple}{P_{ABCD}=2(AB+AD)} } \\ AB=m+6\\ AD=m-2\\ \downarrow\\ P_{ABCD}=2(m+6+m-2) \\ P_{ABCD}=2(2m+4) \\ \boxed{P_{ABCD}=4m+8}$(length units)

Therefore the correct answer is answer C.