The image contains a math problem focused on expanding a square to a rectangle by altering its side lengths. The problem has four parts:

Problem Summary: A square of unknown side length $x$ inches has one side length increased by 4 inches and the other side increased by 7 inches.

Part (a):

You are to label the second diagram representing a new rectangle, constructed by increasing the sides as described. The original square has side lengths $x$, and you extend one side by 4 inches and the other by 7 inches.

Part (b):

Label each portion of the second diagram with their areas in terms of $x$ (where applicable). State the product of $(x + 4)$ and $(x + 7)$ as a trinomial.

Part (c):

If the original square had a side length of $x = 2$ inches, what is the area of the second rectangle? Show how you arrived at your answer.

Part (d):

Verify that the trinomial you found in part (b) has the same value as the area calculated in part (c) for $x = 2$.

Let's go through each part:

(a) Labeling the Diagram:

• Original square has a side length of $x$, so the area is $x^2$.
• The new rectangle's dimensions are $x + 4$ (one side increased by 4 inches) and $x + 7$ (the other side increased by 7 inches).

Thus, the dimensions of the rectangle are $x + 4$ and $x + 7$.

(b) Labeling Areas and Expanding the Trinomial:

The total area of the new rectangle is the product of the two sides: $(x + 4)(x + 7) = x^2 + 7x + 4x + 28 = x^2 + 11x + 28$

Each part of the diagram represents:

1. $x^2$ for the original square.
2. $7x$ for the rectangle formed by extending one side by 7 inches.
3. $4x$ for the rectangle formed by extending the other side by 4 inches.
4. 28 for the small rectangle formed by increasing both sides by 4 and 7 inches.

(c) Area When $x = 2$:

Substitute $x = 2$ into the dimensions of the rectangle: $\text{Length} = 2 + 4 = 6, \quad \text{Width} = 2 + 7 = 9$ The area of the second rectangle is: $\text{Area} = 6 \times 9 = 54 \text{ square inches.}$

(d) Verifying the Trinomial for $x = 2$:

Substitute $x = 2$ into the trinomial $x^2 + 11x + 28$: $(2)^2 + 11(2) + 28 = 4 + 22 + 28 = 54$

The area from the trinomial is also 54, confirming the result from part (c).

Do you need more details on any step, or would you like to go over specific sections further?

Related Questions:

1. What is the significance of expanding the square to a rectangle geometrically?
2. How does the distributive property apply when expanding binomials?
3. Can you explain how each area component represents a part of the expanded rectangle?
4. How does the formula for the area of a rectangle relate to this problem?
5. Why is it important to verify answers algebraically, as done in part (d)?

Tip: When expanding a binomial product, always group like terms to simplify expressions accurately.