Find Rectangle Width: Solving Area = m²+4m-12 with Length m-2

First, let's look at rectangle $ABCD$:

(Drawing - marking the given data about AB on it)

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

$\begin{cases} \textcolor{red}{S_{ABCD}}= m^2+4m-12 \\ \textcolor{blue}{AB}=m-2\\ \end{cases}$

(we'll use colors here for clarity of the solution later)

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

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

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

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

$S_{ABCD}=AB\cdot AD$

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

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

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 side adjacent to the given side in the rectangle (denoted by m), meaning we want to get an expression for the length of side $AD$,

Let's return then to the equation we reached before and isolate $AD$, we'll do this by dividing both sides of the equation by the algebraic expression that is the coefficient of $AD$, which is: $(m-2)$:

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

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

$m^2+4m-12$

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

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

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

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

In the final stage, after we factored the numerator of the fraction and reduced the fraction,

(Drawing - with the found AD length)

Therefore the correct answer is answer C.