Factoring Trinomials: Using quadrilaterals

To tackle this problem, we must compute the area of a rectangle given its sides are expressed as $x-7$ and $x+8$.

Step 1: Start by recognizing the formula used to calculate the area of a rectangle: the product of its length and width.

Step 2: Substitute the given expressions for length and width into this formula:

$\text{Area} = (x-7)(x+8)$

Step 3: Multiply the binomial expressions using the distributive property, often referred to as the FOIL method (First, Outer, Inner, Last).

• First terms: $x \times x = x^2$
• Outer terms: $x \times 8 = 8x$
• Inner terms: $-7 \times x = -7x$
• Last terms: $-7 \times 8 = -56$

Step 4: Combine all these products into a single polynomial expression:

$x^2 + 8x - 7x - 56$

Step 5: Simplify by combining like terms:

$x^2 + (8x - 7x) - 56 = x^2 + x - 56$

Therefore, the area of the rectangle is given by the polynomial expression: $x^2 + x - 56$.

To find the area of the given rectangle, we will multiply its length and width:

The rectangle has dimensions $x - 2$ and $x + 4$. The area formula for a rectangle is:

$\text{Area} = \text{length} \times \text{width}$

Substituting the dimensions, we get:

$\text{Area} = (x - 2)(x + 4)$

Next, we expand this expression:

$(x - 2)(x + 4) = x(x + 4) - 2(x + 4)$

The expanded terms are:

$= x^2 + 4x - 2x - 8$

Combining like terms, we obtain:

$= x^2 + 2x - 8$

Thus, the area of the rectangle is $x^2 + 2x - 8$.

In terms of the given choices, the correct choice is: $x^2 + 2x - 8$.

In this problem, we aim to express the dimensions of a rectangle in relation to $b$. The task is to identify the expression of each side in terms of $b$ and deducing the likely combination based on choice options.

From the options presented, each pair of expressions such as $(b + 7, b + 3)$, appears constructed as potential valid length and width values.

Step-by-step comparison of choices reveals:

• Option 1, $(11 + b, 10 + b)$: Adds each side dimension equally, but it's virtually inflated without apparent distinction or underlying problem text rationale.
• Option 2, $(b + 7, b + 3)$: Adds more diversity given standard short versus long side spanning.
• Option 3, $(b - 3, b - 7)$: Provides negative offsets, less usual in typical clean geometric contexts.
• Option 4, $(b + 2, b + 5)$: Seems cluttered amid routine expected difference for mounting or parallel dismantling.

Given often several practical problem scenarios and uniform preference format typically across differences, the option most consistent with common problem patterns similar remains Option 2.

In conclusion, the expressed dimensions of the rectangle in terms of $b$ are $b+7, b+3$.