`(x + 2) (x + 1) = (x -2) (x -3)`
` (y+3)(y+2)=(y-1)(y-2)`

Options

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

Explanation

To solve the equations step by step, we will start with the first equation and then move on to the second one.

### Step 1: Solve the first equation
The first equation is:
\[ (x + 2)(x + 1) = (x - 2)(x - 3) \]

**Expanding both sides:**
- Left side:
\[ (x + 2)(x + 1) = x^2 + x + 2x + 2 = x^2 + 3x + 2 \]
- Right side:
\[ (x - 2)(x - 3) = x^2 - 3x - 2x + 6 = x^2 - 5x + 6 \]

Now we have:
\[ x^2 + 3x + 2 = x^2 - 5x + 6 \]

**Step 2: Simplify the equation**
Subtract \(x^2\) from both sides:
\[ 3x + 2 = -5x + 6 \]

**Step 3: Combine like terms**
Add \(5x\) to both sides:
\[ 3x + 5x + 2 = 6 \]
\[ 8x + 2 = 6 \]

**Step 4: Isolate \(x\)**
Subtract 2 from both sides:
\[ 8x = 4 \]
Now divide by 8:
\[ x = \frac{4}{8} = \frac{1}{2} = 0.5 \]

### Step 5: Solve the second equation
The second equation is:
\[ (y + 3)(y + 2) = (y - 1)(y - 2) \]

**Expanding both sides:**
- Left side:
\[ (y + 3)(y + 2) = y^2 + 2y + 3y + 6 = y^2 + 5y + 6 \]
- Right side:
\[ (y - 1)(y - 2) = y^2 - 2y - y + 2 = y^2 - 3y + 2 \]

Now we have:
\[ y^2 + 5y + 6 = y^2 - 3y + 2 \]

**Step 6: Simplify the equation**
Subtract \(y^2\) from both sides:
\[ 5y + 6 = -3y + 2 \]

**Step 7: Combine like terms**
Add \(3y\) to both sides:
\[ 5y + 3y + 6 = 2 \]
\[ 8y + 6 = 2 \]

**Step 8: Isolate \(y\)**
Subtract 6 from both sides:
\[ 8y = -4 \]
Now divide by 8:
\[ y = \frac{-4}{8} = \frac{-1}{2} = -0.5 \]

### Step 9: Determine the relationship between \(x\) and \(y\)
Now we have:
- \(x = 0.5\)
- \(y = -0.5\)

Since \(0.5 > -0.5\), we conclude that:
\[ x > y \]

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