Which expression is equivalent to \( \left(\frac{3}{2} p+1\right)\left(\frac{1}{2} p+3\right) \) ? A. \( 2 p^{2}+3 \) B. \( 4 p^{2}+3 \) C. \( \frac{3}{4} p^{2}+5 p+3 \) D. \( \frac{3}{4} p^{2}+10 p+3 \)

To find the equivalent expression for \( \left(\frac{3}{2} p+1\right)\left(\frac{1}{2} p+3\right) \), we can expand it using the distributive property (also known as the FOIL method for binomials). 1. **Multiply the first terms**: \[ \frac{3}{2} p \cdot \frac{1}{2} p = \frac{3}{4} p^2 \] 2. **Multiply the outer terms**: \[ \frac{3}{2} p \cdot 3 = \frac{9}{2} p \] 3. **Multiply the inner terms**: \[ 1 \cdot \frac{1}{2} p = \frac{1}{2} p \] 4. **Multiply the last terms**: \[ 1 \cdot 3 = 3 \] Now, we combine all these results: \[ \frac{3}{4} p^2 + \frac{9}{2} p + \frac{1}{2} p + 3 \] Combine the \( p \) terms: \[ \frac{9}{2} p + \frac{1}{2} p = \frac{10}{2} p = 5 p \] So we have: \[ \frac{3}{4} p^2 + 5p + 3 \] Thus, the final equivalent expression is: \[ \frac{3}{4} p^2 + 5p + 3 \] The correct answer is **C. \( \frac{3}{4} p^{2}+5 p+3 \)**.