I. `x^(2)+5x+6=0`
II. ` y^(2) + 7y + 12 = 0`

Options

A. if ` x ge y,`i.e., x is greater than or equal to y.
B. if `x gt y`, i.e., x is greater than y.
C. if ` x le y`, i.e., x is less than or equal to y.
D. if ` x lt y`, i.e., x is less than y.

Explanation

To solve the equations step by step, we will tackle each equation separately.

### Step 1: Solve the first equation \( x^2 + 5x + 6 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = 5 \), and \( c = 6 \).
2. **Factor the quadratic**: We need to find two numbers that multiply to \( c \) (which is 6) and add up to \( b \) (which is 5). The numbers 2 and 3 satisfy this condition.
3. **Rewrite the equation**: We can express the quadratic as:
\[ (x + 2)(x + 3) = 0 \]
4. **Set each factor to zero**:
\[ x + 2 = 0 \quad \text{or} \quad x + 3 = 0 \]
5. **Solve for \( x \)**:
\[ x = -2 \quad \text{or} \quad x = -3 \]

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

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = 7 \), and \( c = 12 \).
2. **Factor the quadratic**: We need to find two numbers that multiply to \( c \) (which is 12) and add up to \( b \) (which is 7). The numbers 3 and 4 satisfy this condition.
3. **Rewrite the equation**: We can express the quadratic as:
\[ (y + 3)(y + 4) = 0 \]
4. **Set each factor to zero**:
\[ y + 3 = 0 \quad \text{or} \quad y + 4 = 0 \]
5. **Solve for \( y \)**:
\[ y = -3 \quad \text{or} \quad y = -4 \]

### Step 3: Compare the values of \( x \) and \( y \)

From the solutions:
- The values of \( x \) are \( -2 \) and \( -3 \).
- The values of \( y \) are \( -3 \) and \( -4 \).

### Step 4: Determine the relationship between \( x \) and \( y \)

We need to check if \( x \) is greater than or equal to \( y \):
- If \( x = -2 \), then \( -2 \geq -3 \) (true).
- If \( x = -3 \), then \( -3 \geq -4 \) (true).

Thus, in both cases, \( x \) is greater than or equal to \( y \).

### Final Answer:
The values of \( x \) are \( -2 \) and \( -3 \), and the values of \( y \) are \( -3 \) and \( -4 \). The relationship \( x \geq y \) holds true.

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