We are given that the area of the rectangle is $8x^2 - 32$ and the length is $x - 2$. The task is to determine which of the provided expressions could represent the width of the rectangle.

Step 1: Area Formula for a Rectangle

The area of a rectangle is calculated using the formula: $\text{Area} = \text{Length} \times \text{Width}$ Here, the given area is $8x^2 - 32$, and the length is $x - 2$. Let the width be $W$. So: $\text{Area} = (\text{Length}) \times (\text{Width})$ This gives: $8x^2 - 32 = (x - 2) \times W$

Step 2: Solving for the Width

To find $W$, divide both sides of the equation by $x - 2$: $W = \frac{8x^2 - 32}{x - 2}$

Now, simplify the expression for the width.

Step 3: Factor the Numerator

We can factor the numerator $8x^2 - 32$ as follows: $8x^2 - 32 = 8(x^2 - 4)$ Notice that $x^2 - 4$ is a difference of squares, so it can be factored further: $x^2 - 4 = (x - 2)(x + 2)$ Thus, the numerator becomes: $8(x - 2)(x + 2)$

Step 4: Simplify the Expression

Now, substitute this factored form into the equation for the width: $W = \frac{8(x - 2)(x + 2)}{x - 2}$ Since $x - 2$ appears in both the numerator and denominator, we can cancel it out, assuming $x \neq 2$: $W = 8(x + 2)$

Step 5: Identify the Correct Option

We now see that the width of the rectangle is $8(x + 2)$. However, none of the given options is exactly $8(x + 2)$, but we can factor out a common factor of 4 from the width expression: $W = 4 \times 2(x + 2) = 4(x + 2)$ This matches option A.

Final Answer:

The correct answer is A. $4(x + 2)$.

Would you like more details or have any questions about this explanation?

Here are 5 related questions for further practice:

1. If the width of the rectangle was given as $4(x + 3)$, what would be the area?
2. What is the geometric significance of factoring the quadratic expression in the area formula?
3. How does the distributive property help in simplifying expressions in geometry problems?
4. What would happen if the given expression for the area was not factorizable?
5. How would the problem change if the length of the rectangle were a different linear expression?

Tip: Always check for common factors in algebraic expressions before attempting more complicated operations like division. It simplifies the problem!