For the two given questions I and II
`3p^2 - 15p-18 = 0`
`q^2 + 7q + 12 = 0`

Options

A. if p is greater than q.
B. if p is smaller than q
C. if p is equal to q
D. if p is either equal to or smaller than q

Answer

To solve the given equations and determine the relationship between the values of \( p \) and \( q \), we will follow these steps:

### Step 1: Solve the first equation \( 3p^2 - 15p - 18 = 0 \)

1. **Factor the equation**:
We can factor the quadratic equation by first simplifying it. Divide all terms by 3:
\[ p^2 - 5p - 6 = 0 \]

2. **Finding factors**:
We need two numbers that multiply to \(-6\) (the constant term) and add up to \(-5\) (the coefficient of \( p \)). The numbers \(-6\) and \(1\) satisfy this condition:
\[ (p - 6)(p + 1) = 0 \]

3. **Setting each factor to zero**:
\[ p - 6 = 0 \quad \Rightarrow \quad p = 6 \]
\[ p + 1 = 0 \quad \Rightarrow \quad p = -1 \]

### Step 2: Solve the second equation \( q^2 + 7q + 12 = 0 \)

1. **Factor the equation**:
We need two numbers that multiply to \(12\) and add up to \(7\). The numbers \(3\) and \(4\) satisfy this condition:
\[ (q + 3)(q + 4) = 0 \]

2. **Setting each factor to zero**:
\[ q + 3 = 0 \quad \Rightarrow \quad q = -3 \]
\[ q + 4 = 0 \quad \Rightarrow \quad q = -4 \]

### Step 3: Compare the values of \( p \) and \( q \)

We have the following values:
- For \( p \): \( 6 \) and \( -1 \)
- For \( q \): \( -3 \) and \( -4 \)

### Step 4: Determine the relationship

Comparing the values:
- \( 6 > -3 \)
- \( 6 > -4 \)
- \( -1 > -3 \)
- \( -1 > -4 \)

Thus, both values of \( p \) are greater than both values of \( q \).

### Conclusion

The relationship is that \( p \) is greater than \( q \).