To determine the side lengths of a rectangle represented by the polynomial [tex]\( x^2 - 4x - 12 \)[/tex], we need to factorize the polynomial and identify which factors correspond to the given options.
1. Factorize the polynomial [tex]\( x^2 - 4x - 12 \)[/tex]:
We need to express the polynomial as a product of two binomials. There are several methods to factor polynomials, such as factoring by grouping, using the quadratic formula to identify the roots, or recognizing common patterns. However, for quadratics like this, we often look for two numbers that multiply to give the constant term (-12) and add to give the linear coefficient (-4).
2. Identify the two numbers:
Let's list pairs of factors of -12 and see which pair adds up to -4:
- (1, -12), (2, -6), (-2, 6), (3, -4), (-3, 4)
The pair that adds up to -4 is (2, -6).
3. Write the polynomial as a product of factors:
Given our solution, we can write the polynomial as:
[tex]\[
x^2 - 4x - 12 = (x + 2)(x - 6)
\][/tex]
4. Compare to the given options:
The factorization yields the side lengths:
[tex]\[
(x + 2) \quad \text{and} \quad (x - 6)
\][/tex]
5. Match the factorized form with one of the options:
Given the options:
- [tex]\((x+2)\)[/tex] and [tex]\((x-6)\)[/tex]
- [tex]\((x+6)\)[/tex] and [tex]\((x-2)\)[/tex]
- [tex]\((x+2)\)[/tex] and [tex]\((x-10)\)[/tex]
- [tex]\((x+10)\)[/tex] and [tex]\((x-2)\)[/tex]
The side lengths that match our factorization are [tex]\((x + 2)\)[/tex] and [tex]\((x - 6)\)[/tex].
Therefore, the side lengths that should be used to model the rectangle are [tex]\((x+2)\)[/tex] and [tex]\((x-6)\)[/tex].
The correct answer is the first option:
[tex]\[
(x+2) \quad \text{and} \quad (x-6)
\][/tex]
