Let's break this problem into two parts:

Part a: Write an expression that represents the perimeter of the polygon.

The polygon has 5 sides, and the length of each side is given by different expressions:

• $2x$
• $2x + 4$
• $3x$
• $x + 5$
• $3x - 1$

The perimeter is the sum of the lengths of all sides, so we need to add these expressions together:

$\text{Perimeter} = 2x + (2x + 4) + 3x + (x + 5) + (3x - 1)$

Now, combine like terms:

$\text{Perimeter} = (2x + 2x + 3x + x + 3x) + (4 + 5 - 1)$ $\text{Perimeter} = 11x + 8$

Part b: If the perimeter of the polygon is 74 units, what is the length of each side?

We are given that the perimeter is 74 units, so we can set up the equation:

$11x + 8 = 74$

To solve for $x$, subtract 8 from both sides:

$11x = 66$

Now, divide by 11:

$x = 6$

Length of Each Side:

• $2x = 2(6) = 12$
• $2x + 4 = 2(6) + 4 = 12 + 4 = 16$
• $3x = 3(6) = 18$
• $x + 5 = 6 + 5 = 11$
• $3x - 1 = 3(6) - 1 = 18 - 1 = 17$

So, the lengths of the sides are 12, 16, 18, 11, and 17 units.

Would you like more details on any step?

Here are 5 related questions to consider:

1. How would the perimeter expression change if the polygon had 6 sides?
2. Can you generalize the perimeter formula for any polygon with variable side lengths?
3. What happens to the perimeter if the value of $x$ doubles?
4. How would you find the area if it were a regular polygon?
5. Can you use this perimeter approach to find the sides of other polygons like hexagons or triangles?

Tip: Always simplify expressions step-by-step when combining like terms to avoid errors!