I. ` x^(2) + 4x+4 = 0`
II. ` y^(2) - 8y + 16 = 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 given equations step by step, we will first focus on each equation separately and then find the relationship between the variables \( x \) and \( y \).

### Step 1: Solve the first equation \( x^2 + 4x + 4 = 0 \)

1. **Identify the equation**: We have \( x^2 + 4x + 4 = 0 \).
2. **Factor the equation**: Notice that this can be factored as \( (x + 2)^2 = 0 \).
3. **Set the factor equal to zero**:
\[ x + 2 = 0 \]
4. **Solve for \( x \)**:
\[ x = -2 \]

### Step 2: Solve the second equation \( y^2 - 8y + 16 = 0 \)

1. **Identify the equation**: We have \( y^2 - 8y + 16 = 0 \).
2. **Factor the equation**: This can be factored as \( (y - 4)^2 = 0 \).
3. **Set the factor equal to zero**:
\[ y - 4 = 0 \]
4. **Solve for \( y \)**:
\[ y = 4 \]

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

1. **We have found**:
- \( x = -2 \)
- \( y = 4 \)
2. **Compare \( x \) and \( y \)**:
\[ y > x \quad \text{(since \( 4 > -2 \))} \]

### Conclusion

Thus, we can conclude that \( x < y \).

### Final Answer
The relationship between \( x \) and \( y \) is \( x < y \).

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