The problem involves a 5 inch by 7 inch photograph placed inside a picture frame, where both the length and width of the frame are $2x$ inches larger than the photograph’s dimensions. We are asked to find an expression representing the perimeter of the frame.

Step-by-step breakdown:

1. Dimensions of the photograph:

   • Width of the photograph: 5 inches
   • Length of the photograph: 7 inches

2. Dimensions of the frame:

   • Since the frame's width and length are $2x$ inches larger than the photograph, the new dimensions of the frame are:
     • Width of the frame: $5 + 2x$ inches
     • Length of the frame: $7 + 2x$ inches

3. Perimeter of the frame: The formula for the perimeter $P$ of a rectangle is: $P = 2 \times (\text{Length} + \text{Width})$ Substituting the values: $P = 2 \times \left((7 + 2x) + (5 + 2x)\right)$ Simplify the expression inside the parentheses: $P = 2 \times \left(12 + 4x\right)$ Distribute the 2: $P = 24 + 8x$

So, the expression for the perimeter of the frame is $8x + 24$.

The correct answer is:

$\boxed{8x + 24}$

Do you want further details or have any questions?

Here are 5 related questions to expand on this topic:

1. How would the perimeter change if the frame’s dimensions increased by $3x$ instead of $2x$?
2. Can you express the area of the frame in terms of $x$?
3. What would the perimeter be if $x = 3$?
4. How would you calculate the perimeter if the photo were circular instead of rectangular?
5. How do you factor the expression $8x + 24$ to make it more simplified?

Tip: When dealing with perimeter problems, always remember that you are adding twice the length and twice the width!