In each questions two equations numbered I. and II. are given. You have to solve both the equations and mark appropriate answer
I. ` 4 x^(2) + 27 x + 45 = 0 `
II. ` 5 y^(2) + 42 y + 88 = 0 `

Options

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

Answer

To solve the given equations, we will follow these steps:

### Step 1: Solve Equation I
The first equation is:
\[ 4x^2 + 27x + 45 = 0 \]

To solve this quadratic equation, we can use the factorization method. We need to find two numbers that multiply to \(4 \times 45 = 180\) and add up to \(27\).

1. The factors of \(180\) that add up to \(27\) are \(12\) and \(15\).
2. Rewrite the equation:
\[ 4x^2 + 12x + 15x + 45 = 0 \]
3. Group the terms:
\[ (4x^2 + 12x) + (15x + 45) = 0 \]
4. Factor out the common terms:
\[ 4x(x + 3) + 15(x + 3) = 0 \]
5. Factor out \((x + 3)\):
\[ (x + 3)(4x + 15) = 0 \]
6. Set each factor to zero:
- \(x + 3 = 0 \Rightarrow x = -3\)
- \(4x + 15 = 0 \Rightarrow 4x = -15 \Rightarrow x = -\frac{15}{4} = -3.75\)

### Step 2: Solve Equation II
The second equation is:
\[ 5y^2 + 42y + 88 = 0 \]

Similarly, we will factor this equation. We need to find two numbers that multiply to \(5 \times 88 = 440\) and add up to \(42\).

1. The factors of \(440\) that add up to \(42\) are \(20\) and \(22\).
2. Rewrite the equation:
\[ 5y^2 + 20y + 22y + 88 = 0 \]
3. Group the terms:
\[ (5y^2 + 20y) + (22y + 88) = 0 \]
4. Factor out the common terms:
\[ 5y(y + 4) + 22(y + 4) = 0 \]
5. Factor out \((y + 4)\):
\[ (y + 4)(5y + 22) = 0 \]
6. Set each factor to zero:
- \(y + 4 = 0 \Rightarrow y = -4\)
- \(5y + 22 = 0 \Rightarrow 5y = -22 \Rightarrow y = -\frac{22}{5} = -4.4\)

### Step 3: Compare the Solutions
Now we have the solutions:
- From Equation I: \(x = -3\) and \(x = -3.75\)
- From Equation II: \(y = -4\) and \(y = -4.4\)

We will compare these values:
1. Compare \(x = -3\) with \(y = -4\):
- \(-3 > -4\) (So, \(x > y\))
2. Compare \(x = -3.75\) with \(y = -4.4\):
- \(-3.75 > -4.4\) (So, \(x > y\))

### Conclusion
In both cases, we find that \(x > y\).

### Final Answer
The relationship between \(x\) and \(y\) is:
\[ x > y \]