I. `p^(2)+5p+6 = 0`
II. ` q^(2) + 3q + 2 = 0`

Options

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

Explanation

To solve the given quadratic equations and determine the relationship between their solutions, we will follow these steps:

### Step 1: Solve the first equation `p^2 + 5p + 6 = 0`.

To solve this quadratic equation, we can factor it. We need to find two numbers that multiply to 6 (the constant term) and add up to 5 (the coefficient of the linear term).

The factors of 6 that add up to 5 are 2 and 3. Thus, we can write:

\[ p^2 + 5p + 6 = (p + 2)(p + 3) = 0 \]

Setting each factor to zero gives us:

\[ p + 2 = 0 \quad \Rightarrow \quad p = -2 \]
\[ p + 3 = 0 \quad \Rightarrow \quad p = -3 \]

### Step 2: Solve the second equation `q^2 + 3q + 2 = 0`.

Similarly, we will factor this quadratic equation. We need two numbers that multiply to 2 and add up to 3.

The factors of 2 that add up to 3 are 1 and 2. Thus, we can write:

\[ q^2 + 3q + 2 = (q + 1)(q + 2) = 0 \]

Setting each factor to zero gives us:

\[ q + 1 = 0 \quad \Rightarrow \quad q = -1 \]
\[ q + 2 = 0 \quad \Rightarrow \quad q = -2 \]

### Step 3: Compare the values of `p` and `q`.

From our solutions, we have:
- The values of `p` are -2 and -3.
- The values of `q` are -1 and -2.

Now, we can compare the values:
- For \( p = -2 \), \( q = -2 \) (equal).
- For \( p = -3 \), \( q = -1 \) (where \( q > p \)).

### Conclusion: Determine the relationship between `p` and `q`.

From the comparisons:
- The maximum value of `p` (which is -2) is equal to the minimum value of `q` (which is also -2).
- The other value of `p` (-3) is less than the maximum value of `q` (-1).

Thus, we conclude that \( q \) is greater than or equal to \( p \).

### Final Answer:
The relationship between \( p \) and \( q \) is that \( q \geq p \).

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