To determine the expression that represents the perimeter of the given rectangle, let's follow these steps:
1. Identify the expressions given for the width and length of the rectangle:
- Width: [tex]\(k + 3\)[/tex] cm
- Length: [tex]\(k - 9\)[/tex] cm
2. Recall the formula for the perimeter of a rectangle:
[tex]\[
P = 2(\text{width} + \text{length})
\][/tex]
3. Substitute the given expressions for width and length into the perimeter formula:
[tex]\[
P = 2((k + 3) + (k - 9))
\][/tex]
4. Simplify the expression inside the parentheses:
[tex]\[
(k + 3) + (k - 9) = k + k + 3 - 9 = 2k - 6
\][/tex]
5. Multiply the simplified expression by 2:
[tex]\[
P = 2 \times (2k - 6) = 4k - 12
\][/tex]
6. Match the simplified perimeter expression with the provided choices:
- [tex]\( -8 + 4k \)[/tex]
- [tex]\( 8k - 16 \)[/tex]
- [tex]\( 2k - 6 \)[/tex]
- [tex]\( -12 + 4k \)[/tex]
The expression [tex]\(4k - 12\)[/tex] matches the form of the last choice, [tex]\(-12 + 4k\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{-12 + 4k}
\][/tex]
