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.
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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: \[ 2x^2 + 2x + 3x + 3 = 2x(x + 1) + 3(x + 1) = (2x + 3)(x + 1) \]
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Thus, we have: \[ 2x^2 + 5x + 3 = (2x + 3)(x + 1) \]
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Now substituting back into our expression for length: \[ L = \frac{(2x + 3)(x + 1)}{x + 1} \]
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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: \[ L(x) = \text{Fill this with } 2x + 3 \]