Question

Q
29 Which expression is equivalent to (3/2 * p + 1)(1/2 * p + 3)
D 3/4 * p ^ 2 + 10p + 3
A 2p ^ 2 + 3
C 3/4 * p ^ 2 + 5p + 3
B 4p ^ 2 + 3
D

Answer 1

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).

1. **First:** Multiply the first terms:
\[
\frac{3}{2} p \cdot \frac{1}{2} p = \frac{3}{4} p^2
\]

2. **Outer:** Multiply the outer terms:
\[
\frac{3}{2} p \cdot 3 = \frac{9}{2} p
\]

3. **Inner:** Multiply the inner terms:
\[
1 \cdot \frac{1}{2} p = \frac{1}{2} p
\]

4. **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:** \(3/4 p^2 + 5p + 3\).