In each question, two equations numbered I and II are given. You have to solve both the equations and markthe appropriate answer.
I `x^2+14x+45=0`
II `y^2+19y+88=0`

Options

A. `xgey`
B. `xlty`
C. `xley`
D. relation can’t be established

Answer

To solve the given equations step by step, we will first solve each quadratic equation separately and then analyze the results.

### Step 1: Solve the first equation \(x^2 + 14x + 45 = 0\)

1. **Identify the coefficients**:
- Here, \(a = 1\), \(b = 14\), and \(c = 45\).

2. **Factor the quadratic equation**:
- We need to find two numbers that multiply to \(c\) (45) and add up to \(b\) (14).
- The numbers that satisfy this are 5 and 9 because \(5 \times 9 = 45\) and \(5 + 9 = 14\).

3. **Rewrite the equation**:
- We can express the quadratic as:
\[ (x + 5)(x + 9) = 0 \]

4. **Set each factor to zero**:
- \(x + 5 = 0\) or \(x + 9 = 0\)

5. **Solve for \(x\)**:
- From \(x + 5 = 0\), we get \(x = -5\).
- From \(x + 9 = 0\), we get \(x = -9\).

Thus, the solutions for the first equation are:
\[ x = -5 \quad \text{and} \quad x = -9 \]

### Step 2: Solve the second equation \(y^2 + 19y + 88 = 0\)

1. **Identify the coefficients**:
- Here, \(a = 1\), \(b = 19\), and \(c = 88\).

2. **Factor the quadratic equation**:
- We need to find two numbers that multiply to \(c\) (88) and add up to \(b\) (19).
- The numbers that satisfy this are 11 and 8 because \(11 \times 8 = 88\) and \(11 + 8 = 19\).

3. **Rewrite the equation**:
- We can express the quadratic as:
\[ (y + 11)(y + 8) = 0 \]

4. **Set each factor to zero**:
- \(y + 11 = 0\) or \(y + 8 = 0\)

5. **Solve for \(y\)**:
- From \(y + 11 = 0\), we get \(y = -11\).
- From \(y + 8 = 0\), we get \(y = -8\).

Thus, the solutions for the second equation are:
\[ y = -11 \quad \text{and} \quad y = -8 \]

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

- We have the values of \(x\) as \(-5\) and \(-9\).
- We have the values of \(y\) as \(-11\) and \(-8\).

Now we compare:
- For \(x = -5\):
- \(-5\) is greater than both \(-8\) and \(-11\).
- For \(x = -9\):
- \(-9\) is less than \(-8\) but greater than \(-11\).

### Conclusion

Since \(x\) takes two values and \(y\) takes two values, we find:
- When \(x = -5\), \(x > y\) (both values of \(y\)).
- When \(x = -9\), \(x < y\) (for \(y = -8\)) but \(x > y\) (for \(y = -11\)).

Thus, there is no consistent relationship established between \(x\) and \(y\) across all values.

### Final Answer
The correct answer is: **Option 4: No relationship can be established.**