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.