A rectangle with an area of [tex]$x^2-4x-12$[/tex] square units is represented by the model. What side lengths should be used to model the rectangle?

A. [tex]$(x+2)$[/tex] and [tex][tex]$(x-6)$[/tex][/tex]
B. [tex]$(x+6)$[/tex] and [tex]$(x-2)$[/tex]
C. [tex][tex]$(x+2)$[/tex][/tex] and [tex]$(x-10)$[/tex]
D. [tex]$(x+10)$[/tex] and [tex][tex]$(x-2)$[/tex][/tex]



Answer :

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]