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The image you uploaded contains a problem that asks for expressions in terms of \( x \) for the perimeter of various shapes. Here’s how you can solve each one:

### Problem 1: Perimeter Expressions

For each shape, you need to sum the lengths of all sides to find the perimeter expression.

**a) Square**  
Each side of the square is given as \( x + 1 \).  
- **Perimeter**: \( 4(x + 1) \)
- **Simplified**: \( 4x + 4 \)

**b) Rectangle**  
The lengths of the sides are \( 4x + 2 \) and \( 3x \).  
- **Perimeter**: \( 2(4x + 2) + 2(3x) \)
- **Simplified**: \( 8x + 4 + 6x = 14x + 4 \)

**c) Hexagon**  
Each side of the hexagon is given as \( x + 2 \).  
- **Perimeter**: \( 6(x + 2) \)
- **Simplified**: \( 6x + 12 \)

**d) Triangle**  
The sides are \( 3x + 5 \), \( 4x + 2 \), and \( 2x + 1 \).  
- **Perimeter**: \( (3x + 5) + (4x + 2) + (2x + 1) \)
- **Simplified**: \( 3x + 5 + 4x + 2 + 2x + 1 = 9x + 8 \)

### Problem 2: Area Expressions (Not requested but visible in the image)

If you also want to find the area for each shape:

**a) Square**  
Area: \( (x + 1)^2 = x^2 + 2x + 1 \)

**b) Rectangle**  
Area: \( (4x + 2)(3x) = 12x^2 + 6x \)

**c) Hexagon**  
(The problem does not give enough information for area calculation; typically, the area of a regular hexagon would require side length and an appropriate formula.)

**d) Triangle**  
(The area of a triangle would need height, which is not provided.)

Would you like more details or have any questions about these solutions?

### Further Questions:
1. What is the perimeter formula for a general polygon with \( n \) sides, each side length \( s \)?
2. How do you find the area of a regular hexagon if the side length is \( s \)?
3. If one side of a square is doubled, how does the perimeter change?
4. How would you derive the area expression for the triangle if its height were known?
5. What are the general steps for simplifying algebraic expressions?

**Tip:** Always combine like terms and apply distributive property for simplifying expressions effectively.

Write an expression in terms of x for the perimeter of each shape. Write each expression in its simplest form.

Learn how to write expressions for the perimeter of various shapes in terms of x. This problem covers key algebraic concepts, including simplification of expressions and perimeter formulas for squares, rectangles, hexagons, and triangles. Suitable for students in Grades 6-8.

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