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

A rectangle has an area of

$m^2+4m-12$ cm² and a length of $m-2$ cm.

Determine the length of the width of the rectangle:

Observe the rectangle $ABCD$:

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

Let's continue and write down the data from 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 greater clarity)

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:

$S_{ABCD}=AB\cdot AD$

We can insert the previously mentioned data into this expression for area to obtain the equation as shown below:

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

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

Our goal is of course to obtain the algebraic expression for the side adjacent to the given side in the rectangle (denoted by m), meaning we want to obtain an expression for the length of side $AD$,

Let's return then to the equation that we previously obtained and isolate $AD$. This is achieved 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 to simplify the algebraic fraction that we obtained. We can do this easily by factoring the numerator of the fraction:

$m^2+4m-12$

Apply quick trinomial factoring as shown below:

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

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 and reduced the fraction,

Therefore the correct answer is answer C.