The problem provides the dimensions of a rectangle:

• Length = $(4x - 5)$ cm
• Width = $(x + 3)$ cm

The perimeter of the rectangle is 46 cm. We are tasked with calculating the area of the rectangle.

Step 1: Recall the perimeter formula

The perimeter $P$ of a rectangle is given by: $P = 2 \times (\text{Length} + \text{Width})$

Substitute the given length and width expressions into the perimeter formula: $46 = 2 \times \left( (4x - 5) + (x + 3) \right)$

Step 2: Simplify the equation

Simplify the right-hand side: $46 = 2 \times (4x - 5 + x + 3)$ $46 = 2 \times (5x - 2)$ Now divide both sides by 2: $23 = 5x - 2$

Step 3: Solve for $x$

To isolate $x$, add 2 to both sides: $25 = 5x$ Now divide by 5: $x = 5$

Step 4: Find the dimensions of the rectangle

Substitute $x = 5$ into the expressions for the length and width:

• Length = $4(5) - 5 = 20 - 5 = 15$ cm
• Width = $5 + 3 = 8$ cm

Step 5: Calculate the area

The area $A$ of a rectangle is given by: $A = \text{Length} \times \text{Width}$ Substitute the values for length and width: $A = 15 \times 8 = 120 \text{ cm}^2$

Thus, the area of the rectangle is $120 \, \text{cm}^2$.

Would you like further details or have any questions on the solution? Here are some related questions for you:

1. What is the formula for the perimeter of a rectangle?
2. How can you express the dimensions of a rectangle algebraically?
3. What would happen if the perimeter were different, say 50 cm? How would you solve that?
4. How do you calculate the area of other shapes, like triangles or circles?
5. How does solving for a variable work when dealing with algebraic expressions?

Tip: Remember to always simplify equations step by step to avoid confusion when solving for variables.