The image contains a mathematical problem where you're tasked with finding the dimensions of a rectangle given the perimeter. The side lengths of the rectangle are represented by the expressions $2x + 3$ and $x - 5$. The perimeter of the rectangle is given as 32 cm.

Step-by-step Solution:

1. Perimeter Formula for a Rectangle: The formula for the perimeter of a rectangle is: $P = 2 \times (\text{length} + \text{width})$ In this case, the length is $2x + 3$ and the width is $x - 5$.

2. Substitute the given values: Since the perimeter is 32 cm, we can set up the equation: $32 = 2 \times \left( (2x + 3) + (x - 5) \right)$

3. Simplify the equation: First, combine the terms inside the parentheses: $32 = 2 \times (3x - 2)$ Now, distribute the 2: $32 = 6x - 4$

4. Solve for $x$: Add 4 to both sides: $36 = 6x$ Now divide by 6: $x = 6$

Final Step: Substitute the value of $x$ into the expressions for the length and width.

• Length: $2x + 3 = 2(6) + 3 = 12 + 3 = 15$ cm
• Width: $x - 5 = 6 - 5 = 1$ cm

Thus, the dimensions of the rectangle are:

• Length = 15 cm
• Width = 1 cm

Would you like a more detailed explanation or have any other questions? Here are 5 related questions to explore:

1. How would the perimeter change if the expressions for the sides were different?
2. What would the area of the rectangle be with these dimensions?
3. How can you find the perimeter of a square with side length $x + 5$?
4. What happens if the perimeter was 40 cm instead of 32 cm?
5. How would you solve a similar problem if given the area instead of the perimeter?

Tip: When dealing with algebraic expressions in geometric problems, always simplify the expressions before substituting values. This reduces mistakes and makes the equation easier to solve!