I. ` x^(2) +20x + 4 = 50 -25x`
II. ` y^(2) - 10y - 24 = 0`

Options

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

Explanation

To solve the given equations step by step, we will first tackle each equation separately and then find the relationship between the values of \(x\) and \(y\).

### Step 1: Solve the first equation
The first equation is:
\[ x^2 + 20x + 4 = 50 - 25x \]

**Step 1.1:** Rearrange the equation
We will move all terms to one side of the equation:
\[ x^2 + 20x + 25x + 4 - 50 = 0 \]
This simplifies to:
\[ x^2 + 45x - 46 = 0 \]

**Step 1.2:** Factor the quadratic equation
We need to factor the quadratic equation \(x^2 + 45x - 46 = 0\). We look for two numbers that multiply to \(-46\) and add to \(45\). The numbers are \(46\) and \(-1\):
\[ (x + 46)(x - 1) = 0 \]

**Step 1.3:** Find the values of \(x\)
Setting each factor to zero gives us:
1. \(x + 46 = 0 \Rightarrow x = -46\)
2. \(x - 1 = 0 \Rightarrow x = 1\)

So, the values of \(x\) are:
\[ x = -46 \quad \text{and} \quad x = 1 \]

### Step 2: Solve the second equation
The second equation is:
\[ y^2 - 10y - 24 = 0 \]

**Step 2.1:** Factor the quadratic equation
We look for two numbers that multiply to \(-24\) and add to \(-10\). The numbers are \(-12\) and \(2\):
\[ (y - 12)(y + 2) = 0 \]

**Step 2.2:** Find the values of \(y\)
Setting each factor to zero gives us:
1. \(y - 12 = 0 \Rightarrow y = 12\)
2. \(y + 2 = 0 \Rightarrow y = -2\)

So, the values of \(y\) are:
\[ y = 12 \quad \text{and} \quad y = -2 \]

### Step 3: Determine the relationship between \(x\) and \(y\)
Now we have the values:
- \(x = -46\) or \(x = 1\)
- \(y = 12\) or \(y = -2\)

**Step 3.1:** Compare the values
1. If \(x = -46\):
- Compare with \(y = 12\): \(-46 < 12\)
- Compare with \(y = -2\): \(-46 < -2\)

2. If \(x = 1\):
- Compare with \(y = 12\): \(1 < 12\)
- Compare with \(y = -2\): \(1 > -2\)

### Conclusion
From the comparisons:
- In both cases, \(x\) is less than \(y\) when \(y = 12\).
- When \(y = -2\), \(x = 1\) is greater than \(y\).

Thus, the relationship between \(x\) and \(y\) can be summarized as:
- \(x\) is less than \(y\) when \(y = 12\).

### Final Answer
The relationship can be stated as:
- \(x < y\) when \(y = 12\).