To find the perimeter \( P \) of a rectangle given by the side lengths represented by two polynomials, we can use the formula for perimeter:
\[ P = 2 \times (\text{length} + \text{width}) \]
In this case, the lengths of the sides are represented as:
- Length \( L = \frac{3}{2}x + 3 \)
- Width \( W = 3x - \frac{1}{2} \)
Now let's find \( L + W \):
\[ L + W = \left( \frac{3}{2}x + 3 \right) + \left( 3x - \frac{1}{2} \right) \]
Combining the terms:
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Combine the \( x \) terms: \[ \frac{3}{2}x + 3x = \frac{3}{2}x + \frac{6}{2}x = \frac{9}{2}x \]
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Combine the constant terms: \[ 3 - \frac{1}{2} = \frac{6}{2} - \frac{1}{2} = \frac{5}{2} \]
So we have:
\[ L + W = \frac{9}{2}x + \frac{5}{2} \]
Now, we need to calculate the perimeter:
\[ P = 2 \times (L + W) = 2 \times \left( \frac{9}{2}x + \frac{5}{2} \right) \]
Distributing the 2:
\[ P = 2 \times \frac{9}{2}x + 2 \times \frac{5}{2} \] \[ P = 9x + 5 \]
Thus, the expression that represents the perimeter of the rectangle is:
\[ P = 9x + 5 \]
None of the options provided match this, but it appears the expression listed as option A, \( \frac{9}{2}x + \frac{5}{2} \), is not correct as it does not correctly represent the perimeter. The correct perimeter expression is \( 9x + 5 \).