I.` x^(2) + 4x = 21`
II` y^(2) - 6y + 8 = 0`

Options

A. ` x gt y`
B. `y gt x`
C. ` x ge y`
D. x = y or a relationship between x and y cannot be established.

Explanation

Let's solve the equations step by step.

### Step 1: Solve for x in the equation \( x^2 + 4x = 21 \)

First, we need to rearrange the equation to set it to zero:

\[ x^2 + 4x - 21 = 0 \]

### Step 2: Factor the quadratic equation

Now we will factor the quadratic equation. We need to find two numbers that multiply to \(-21\) (the constant term) and add to \(4\) (the coefficient of \(x\)). The numbers \(7\) and \(-3\) satisfy this condition:

\[ x^2 + 7x - 3x - 21 = 0 \]

### Step 3: Group the terms

Next, we group the terms:

\[ (x^2 + 7x) + (-3x - 21) = 0 \]

### Step 4: Factor by grouping

Now, we factor out the common terms:

\[ x(x + 7) - 3(x + 7) = 0 \]

This can be simplified to:

\[ (x - 3)(x + 7) = 0 \]

### Step 5: Solve for x

Setting each factor to zero gives us the possible values for \(x\):

\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]
\[ x + 7 = 0 \quad \Rightarrow \quad x = -7 \]

So, the solutions for \(x\) are:

\[ x = 3 \quad \text{or} \quad x = -7 \]

### Step 6: Solve for y in the equation \( y^2 - 6y + 8 = 0 \)

Now we will solve the second equation. We can factor this as well. We need two numbers that multiply to \(8\) and add to \(-6\). The numbers \(2\) and \(4\) fit this requirement:

\[ y^2 - 4y - 2y + 8 = 0 \]

### Step 7: Group the terms

We group the terms:

\[ (y^2 - 4y) + (-2y + 8) = 0 \]

### Step 8: Factor by grouping

Factoring gives us:

\[ y(y - 4) - 2(y - 4) = 0 \]

This simplifies to:

\[ (y - 2)(y - 4) = 0 \]

### Step 9: Solve for y

Setting each factor to zero gives us the possible values for \(y\):

\[ y - 2 = 0 \quad \Rightarrow \quad y = 2 \]
\[ y - 4 = 0 \quad \Rightarrow \quad y = 4 \]

So, the solutions for \(y\) are:

\[ y = 2 \quad \text{or} \quad y = 4 \]

### Step 10: Analyze the relationship between x and y

Now we have the values:

- \(x = 3\) or \(x = -7\)
- \(y = 2\) or \(y = 4\)

We need to analyze the relationship between \(x\) and \(y\):

- For \(x = 3\): \(y = 2\) (3 > 2) and \(y = 4\) (3 < 4)
- For \(x = -7\): \(y = 2\) (-7 < 2) and \(y = 4\) (-7 < 4)

From this analysis, we can conclude that:

- \(x\) can be greater than \(y\) in one case (when \(x = 3\) and \(y = 2\)).
- \(x\) can be less than \(y\) in other cases (when \(x = -7\) and both values of \(y\)).
- There is no consistent relationship between \(x\) and \(y\).

Thus, we cannot establish a definitive relationship between \(x\) and \(y\).