If `(x+2)^(2)le2-y`. What is the maximum possible value of y?

Answer

To solve the inequality \((x + 2)^2 \leq 2 - y\) and find the maximum possible value of \(y\), we can follow these steps:

### Step 1: Rearranging the Inequality
Start by rearranging the inequality to isolate \(y\):
\[ (x + 2)^2 \leq 2 - y \]
Add \(y\) to both sides and subtract \((x + 2)^2\) from both sides:
\[ y \leq 2 - (x + 2)^2 \]

### Step 2: Define a Function
Let’s define a function \(f(x)\) based on the right side of the inequality:
\[ f(x) = 2 - (x + 2)^2 \]
Now we have:
\[ y \leq f(x) \]

### Step 3: Analyzing the Function
Next, we need to analyze the function \(f(x)\) to find its maximum value. The term \((x + 2)^2\) is a perfect square and is always non-negative. Therefore, the minimum value of \((x + 2)^2\) is \(0\), which occurs when \(x + 2 = 0\) or \(x = -2\).

### Step 4: Finding the Maximum Value of \(f(x)\)
Substituting \(x = -2\) into \(f(x)\):
\[ f(-2) = 2 - (-2 + 2)^2 = 2 - 0 = 2 \]
Since \((x + 2)^2\) is always non-negative, the maximum value of \(f(x)\) occurs at \(x = -2\), yielding:
\[ f(x) \leq 2 \]

### Step 5: Conclusion
From the inequality \(y \leq f(x)\) and the maximum value of \(f(x)\) being \(2\), we conclude:
\[ y \leq 2 \]
Thus, the maximum possible value of \(y\) is:
\[ \boxed{2} \]