I. ` p^(2) + 12p+35 = 0`
II. ` 2q^(2) + 22q+56 = 0`

Options

A. if`p lt q,`
B. if`p gt q,`
C. if` p le q,`
D. if` p ge q,`

Explanation

To solve the equations given in the question step by step, we will tackle each quadratic equation separately.

### Step 1: Solve for \( p \) in the equation \( p^2 + 12p + 35 = 0 \)

1. **Identify the coefficients**:
- \( a = 1 \)
- \( b = 12 \)
- \( c = 35 \)

2. **Factor the quadratic**:
- We need two numbers that multiply to \( 35 \) (the constant term) and add up to \( 12 \) (the coefficient of \( p \)).
- The numbers \( 7 \) and \( 5 \) work because \( 7 \times 5 = 35 \) and \( 7 + 5 = 12 \).

3. **Write the factored form**:
\[ (p + 7)(p + 5) = 0 \]

4. **Set each factor to zero**:
- \( p + 7 = 0 \) → \( p = -7 \)
- \( p + 5 = 0 \) → \( p = -5 \)

### Step 2: Solve for \( q \) in the equation \( 2q^2 + 22q + 56 = 0 \)

1. **Divide the entire equation by 2** to simplify:
\[ q^2 + 11q + 28 = 0 \]

2. **Identify the coefficients**:
- \( a = 1 \)
- \( b = 11 \)
- \( c = 28 \)

3. **Factor the quadratic**:
- We need two numbers that multiply to \( 28 \) and add up to \( 11 \).
- The numbers \( 4 \) and \( 7 \) work because \( 4 \times 7 = 28 \) and \( 4 + 7 = 11 \).

4. **Write the factored form**:
\[ (q + 4)(q + 7) = 0 \]

5. **Set each factor to zero**:
- \( q + 4 = 0 \) → \( q = -4 \)
- \( q + 7 = 0 \) → \( q = -7 \)

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

- The values obtained are:
- \( p = -7 \) or \( p = -5 \)
- \( q = -4 \) or \( q = -7 \)

### Step 4: Determine the relationship between \( p \) and \( q \)

1. **Compare the values**:
- For \( p = -7 \) and \( q = -7 \): \( p = q \)
- For \( p = -5 \) and \( q = -4 \): \( q > p \)

### Conclusion

The relationship between \( p \) and \( q \) is that \( q \) is greater than or equal to \( p \).

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