Let's solve this step-by-step.

Step 1: Determine $x$ using the perimeter information

The problem states that the perimeter of hexagon $ABCDEF$ is 40 cm. Adding up all given side lengths, we get: $AB + BC + CD + DE + EF + FA = 40$ where:

• $AB = 2x + 1$
• $BC = 3x + 4$
• $CD = 2x$
• $DE = 2x$
• $EF = 5x$
• $FA = 3x + 4$

Adding these together: $(2x + 1) + (3x + 4) + (2x) + (2x) + (5x) + (3x + 4) = 40$

Simplify: $17x + 9 = 40$

Subtract 9 from both sides: $17x = 31$

Divide by 17: $x = \frac{31}{17} = 1.82 \, \text{(approximately)}$

Step 2: Substitute $x$ to find side lengths

Now, substitute $x = 1.82$ into each expression to find the lengths of each side:

• $AB = 2(1.82) + 1 = 4.64 \, \text{cm}$
• $BC = 3(1.82) + 4 = 9.46 \, \text{cm}$
• $CD = 2(1.82) = 3.64 \, \text{cm}$
• $DE = 2(1.82) = 3.64 \, \text{cm}$
• $EF = 5(1.82) = 9.1 \, \text{cm}$
• $FA = 3(1.82) + 4 = 9.46 \, \text{cm}$

Step 3: Calculate the area

To find the area of this irregular hexagon, we can split it into two rectangles:

1. Rectangle 1: Formed by sides $AB$ and $AF$.

   • Length = $4.64 \, \text{cm}$
   • Width = $9.1 \, \text{cm}$
   • Area = $4.64 \times 9.1 = 42.22 \, \text{cm}^2$

2. Rectangle 2: Formed by sides $CD$ and $DE$.

   • Length