Evaluate `x^(2)+4x+7` when `x=-2+sqrt(-3)`.

Answer

To evaluate the expression \( x^2 + 4x + 7 \) when \( x = -2 + \sqrt{-3} \), we will follow these steps:

### Step 1: Substitute the value of \( x \)
We start by substituting \( x = -2 + \sqrt{-3} \) into the expression:
\[ x^2 + 4x + 7 = (-2 + \sqrt{-3})^2 + 4(-2 + \sqrt{-3}) + 7 \]

### Step 2: Calculate \( x^2 \)
Now, we calculate \( (-2 + \sqrt{-3})^2 \):
\[ (-2 + \sqrt{-3})^2 = (-2)^2 + 2(-2)(\sqrt{-3}) + (\sqrt{-3})^2 \]
\[ = 4 - 4\sqrt{-3} + (-3) = 4 - 4\sqrt{-3} - 3 = 1 - 4\sqrt{-3} \]

### Step 3: Calculate \( 4x \)
Next, we calculate \( 4(-2 + \sqrt{-3}) \):
\[ 4(-2 + \sqrt{-3}) = 4(-2) + 4(\sqrt{-3}) = -8 + 4\sqrt{-3} \]

### Step 4: Combine all parts
Now we can combine all parts:
\[ x^2 + 4x + 7 = (1 - 4\sqrt{-3}) + (-8 + 4\sqrt{-3}) + 7 \]
Combining the constant terms:
\[ 1 - 8 + 7 = 0 \]
And combining the terms with \( \sqrt{-3} \):
\[ -4\sqrt{-3} + 4\sqrt{-3} = 0 \]

### Step 5: Final result
Thus, we have:
\[ x^2 + 4x + 7 = 0 + 0 = 0 \]

So, the final answer is:
\[ \boxed{0} \]