Question

The area of a desktop in the shape of a rectangle is given by the function f(x)=2x2+5x+3
. The width of the desktop is given by the function g(x)=x+1
. What function represents the length of the desktop?

Enter the missing values.

(1 point)
$$
Remaining Attempts : 3

Answer 1

To determine the length of the desktop, we can use the relationship that the area \( A \) of a rectangle is equal to the product of its length \( L \) and width \( W \).

In this case, we are given:
- The area function \( f(x) = 2x^2 + 5x + 3 \)
- The width function \( g(x) = x + 1 \)

The area can be expressed as:
\[
A = L \times W
\]
where \( L \) is the length and \( W \) is the width.

We can rearrange this equation to solve for the length \( L \):
\[
L = \frac{A}{W} = \frac{f(x)}{g(x)}
\]

Substituting the given functions:
\[
L = \frac{f(x)}{g(x)} = \frac{2x^2 + 5x + 3}{x + 1}
\]

Now we need to simplify this expression. We can perform polynomial long division or factor if possible.

1. Factor \( 2x^2 + 5x + 3 \):
We look for two numbers that multiply to \( 2 \times 3 = 6 \) and add to \( 5 \). These numbers are \( 2 \) and \( 3 \).
We can rewrite:
\[
2x^2 + 2x + 3x + 3 = 2x(x + 1) + 3(x + 1) = (2x + 3)(x + 1)
\]

2. Thus, we have:
\[
2x^2 + 5x + 3 = (2x + 3)(x + 1)
\]

3. Now substituting back into our expression for length:
\[
L = \frac{(2x + 3)(x + 1)}{x + 1}
\]

4. Cancel out \( x + 1 \) (assuming \( x + 1 \neq 0 \)):
\[
L = 2x + 3
\]

Therefore, the function that represents the length of the desktop is:
\[
L(x) = 2x + 3
\]

Now, you can fill in the blank:

The function that represents the length of the desktop is:
\[
L(x) = \text{Fill this with } 2x + 3
\]