To find the expression equivalent to \((\frac{3}{2} p + 1)(\frac{1}{2} p + 3)\), we need to perform the multiplication using the distributive property (also known as the FOIL method for binomials).
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First: Multiply the first terms: \[ \frac{3}{2} p \cdot \frac{1}{2} p = \frac{3}{4} p^2 \]
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Outer: Multiply the outer terms: \[ \frac{3}{2} p \cdot 3 = \frac{9}{2} p \]
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Inner: Multiply the inner terms: \[ 1 \cdot \frac{1}{2} p = \frac{1}{2} p \]
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Last: Multiply the last terms: \[ 1 \cdot 3 = 3 \]
Now, combine all these results together: \[ \frac{3}{4} p^2 + \frac{9}{2} p + \frac{1}{2} p + 3 \]
To combine the \(p\) terms, convert \(\frac{9}{2} p\) and \(\frac{1}{2} p\) to a common denominator: \[ \frac{9}{2} p + \frac{1}{2} p = \frac{10}{2} p = 5p \]
Now we can write our expression as: \[ \frac{3}{4} p^2 + 5p + 3 \]
Now, let's look at the answer choices:
- A: \(2p^2 + 3\)
- B: \(4p^2 + 3\)
- C: \(3/4 p^2 + 5p + 3\)
- D: \( \text{(not provided)} \)
The equivalent expression we derived is: \[ \frac{3}{4} p^2 + 5p + 3 \]
Thus, the correct choice is C: \(3/4 p^2 + 5p + 3\).