Area of the Square: Using short multiplication formulas

To find the area of a square with side length $x + 1$, we apply the formula for the area of a square, which is side squared. This means we need to calculate $(x + 1)^2$.

Here are the steps to solve the problem:

• Step 1: Identify the expression for the side length. The side length of the square is given as $x + 1$.
• Step 2: Use the formula for the area of a square: $(\text{side})^2$.
• Step 3: Substitute the side length with $x + 1$: $(x + 1)^2$.
• Step 4: Expand the expression using the formula for the square of a sum: $(a + b)^2 = a^2 + 2ab + b^2$, where $a = x$ and $b = 1$.
• Step 5: Calculation:
• $(x + 1)^2 = x^2 + 2(x)(1) + 1^2$
• $(x + 1)^2 = x^2 + 2x + 1$

Therefore, the algebraic expression for the area of the square is $x^2 + 2x + 1$.

To determine which shape has a larger area, we need to compare the areas of the square and the rectangle:

• Step 1: Calculate the area of the original square:

The side length of the square is $X$, so its area is given by:

$\text{Area of square} = X^2$

• Step 2: Calculate the area of the rectangle:

The dimensions of the rectangle are $X + 3$ cm and $X - 3$ cm. Thus, its area is:

$\text{Area of rectangle} = (X + 3)(X - 3)$

Using the difference of squares formula, we find:

$(X + 3)(X - 3) = X^2 - 9$

• Step 3: Compare the areas:

We compute the difference between the square's area and the rectangle's area:

$X^2 - (X^2 - 9) = 9$

Since 9 is positive, the area of the square is larger than the area of the rectangle.

Therefore, the square has a larger area than the rectangle.