To find the side lengths of the rectangle with an area given by [tex]\( x^2 - 4x - 12 \)[/tex] square units, we need to factorize the quadratic expression.
Here is the step-by-step solution:
1. Given Expression:
[tex]\[
x^2 - 4x - 12
\][/tex]
2. Factorize the Quadratic Expression:
The quadratic expression can be written in the form [tex]\((x - a)(x + b)\)[/tex].
3. Identify the Factors:
The expression [tex]\( x^2 - 4x - 12 \)[/tex] factors into:
[tex]\[
(x - 6)(x + 2)
\][/tex]
By writing the expression [tex]\( x^2 - 4x - 12 \)[/tex] in its factored form [tex]\( (x - 6)(x + 2) \)[/tex], we can determine the side lengths of the rectangle.
4. Side Lengths:
The side lengths of the rectangle are:
[tex]\[
(x - 6) \quad \text{and} \quad (x + 2)
\][/tex]
Therefore, among the given options, the correct side lengths that should be used to model the rectangle are:
[tex]\[
(x + 2) \quad \text{and} \quad (x - 6)
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{(x + 2) \text{ and } (x - 6)}
\][/tex]
