The expression for the area of a rectangle is (4x^2+8x) square units. Find an expression for it's width if the length is (x+2) units.

1. Break Down the Problem

We know the area of a rectangle is given by the formula: $\text{Area} = \text{Length} \times \text{Width}$ We need to find the expression for the width if the area is given as $4x^2 + 8x$ square units and the length is given as $x + 2$ units.

2. Relevant Concepts

We will rearrange the area formula to isolate the width: $\text{Width} = \frac{\text{Area}}{\text{Length}}$

3. Analysis and Detail

Given:

• Area = $4x^2 + 8x$
• Length = $x + 2$

Substituting these values into the formula for width, we have: $\text{Width} = \frac{4x^2 + 8x}{x + 2}$

Next, we will simplify the expression for width using polynomial long division or factoring.

4. Verify and Summarize

We can factor the numerator: $4x^2 + 8x = 4x(x + 2)$ Now substituting back into the width expression: $\text{Width} = \frac{4x(x + 2)}{x + 2}$ Provided $x + 2 \neq 0$, we can cancel $x + 2$: $\text{Width} = 4x$

Final Answer

The expression for the width of the rectangle is $4x$ units.