To solve the problem, we need to find the algebraic expression for the area of the composite shape.

Step 1: Analyze the shape

The given shape is a rectangle that has a smaller section cut out from one corner. To compute the area of the entire shape, we can break it into two parts:

1. A large rectangle at the top
2. A smaller rectangle that needs to be subtracted.

Step 2: Find the area of the large rectangle

The large rectangle has:

• Length = $7x + 1$
• Width = $5x$

So, the area of the large rectangle is: $\text{Area of large rectangle} = (7x + 1)(5x)$

Step 3: Find the area of the smaller rectangle

The smaller rectangle has:

• Length = $4x$
• Width = $3x + 1$

So, the area of the smaller rectangle is: $\text{Area of smaller rectangle} = 4x(3x + 1)$

Step 4: Find the area of the composite shape

The area of the composite shape is the area of the large rectangle minus the area of the smaller rectangle: $\text{Total area} = (7x + 1)(5x) - 4x(3x + 1)$

Step 5: Expand the expressions

Now, we expand each part:

• $(7x + 1)(5x) = 35x^2 + 5x$
• $4x(3x + 1) = 12x^2 + 4x$

Now, subtract the areas: $\text{Total area} = (35x^2 + 5x) - (12x^2 + 4x)$ $= 35x^2 + 5x - 12x^2 - 4x$ $= 23x^2 + x$

Step 6: Check the options

Looking at the options:

• Option A: $31x^2 + 5x$
• Option B: $23x^2 + 2$
• Option C: $35x^2 + 5x$
• Option D: $36x^2$

None of these exactly matches $23x^2 + x$, suggesting that there may be a mistake in the question or options. Please check the problem for accuracy or revalidate the algebraic steps.

Would you like to go over the steps again or ask for additional clarification?

Here are some related questions for further exploration:

1. How do you find the area of a composite shape that includes rectangles and squares?
2. How do you factor a polynomial expression for areas?
3. What would happen if the smaller section was a triangle instead of a rectangle?
4. How do you expand binomials like $(a + b)(c + d)$?
5. How do you simplify an algebraic expression like $35x^2 + 5x - 12x^2 - 4x$?

Tip: Always simplify and combine like terms when working with algebraic expressions.