I. ` x ^(2) + 5x + 6 = 0`
II. `y^(2) + 3y +2 = 0`

Options

A. if ` x gt y`
B. if `x ge y`
C. if ` x lt y`
D. if ` x le y`

Explanation

To solve the given equations and find the relationship between \( x \) and \( y \), we will follow these steps:

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

1. **Identify the quadratic equation**: The equation is in the standard form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 5 \), and \( c = 6 \).
2. **Factor the quadratic**: We need to factor the equation. We look for two numbers that multiply to \( c \) (6) and add to \( b \) (5). The numbers are 2 and 3.
\[ x^2 + 5x + 6 = (x + 2)(x + 3) = 0 \]
3. **Set each factor to zero**:
\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]
4. **Solutions for \( x \)**: The solutions are \( x = -2 \) and \( x = -3 \).

### Step 2: Solve the second equation \( y^2 + 3y + 2 = 0 \)

1. **Identify the quadratic equation**: The equation is in the standard form \( ay^2 + by + c = 0 \), where \( a = 1 \), \( b = 3 \), and \( c = 2 \).
2. **Factor the quadratic**: We look for two numbers that multiply to \( c \) (2) and add to \( b \) (3). The numbers are 1 and 2.
\[ y^2 + 3y + 2 = (y + 1)(y + 2) = 0 \]
3. **Set each factor to zero**:
\[ y + 1 = 0 \quad \Rightarrow \quad y = -1 \]
\[ y + 2 = 0 \quad \Rightarrow \quad y = -2 \]
4. **Solutions for \( y \)**: The solutions are \( y = -1 \) and \( y = -2 \).

### Step 3: Determine the relationship between \( x \) and \( y \)

1. **List the values**:
- From the first equation, \( x \) can be \( -2 \) or \( -3 \).
- From the second equation, \( y \) can be \( -1 \) or \( -2 \).
2. **Compare the values**:
- If \( x = -2 \), then \( y = -2 \) (which means \( x = y \)).
- If \( x = -3 \), then \( y = -1 \) (which means \( x < y \)).
3. **Conclusion**: The relationship can be summarized as:
- \( x \) can be equal to \( y \) when both are \( -2 \).
- \( x \) is less than \( y \) when \( x = -3 \) and \( y = -1 \).

### Final Result:
The relationship between \( x \) and \( y \) can be expressed as:
- \( x \leq y \)

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