If `x=2+sqrt(3)`, then find the value of `x^(4)-4x^(3)+x^(2)+x+1`.

Explanation

To solve the problem where \( x = 2 + \sqrt{3} \) and we need to find the value of \( x^4 - 4x^3 + x^2 + x + 1 \), we will follow these steps:

### Step 1: Calculate \( x^2 \)
We start by calculating \( x^2 \):
\[ x^2 = (2 + \sqrt{3})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 \]
\[ = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

### Step 2: Calculate \( x^3 \)
Next, we calculate \( x^3 \) using \( x^2 \):
\[ x^3 = x^2 \cdot x = (7 + 4\sqrt{3})(2 + \sqrt{3}) \]
Using the distributive property:
\[ = 7 \cdot 2 + 7 \cdot \sqrt{3} + 4\sqrt{3} \cdot 2 + 4\sqrt{3} \cdot \sqrt{3} \]
\[ = 14 + 7\sqrt{3} + 8\sqrt{3} + 12 = 26 + 15\sqrt{3} \]

### Step 3: Calculate \( x^4 \)
Now, we calculate \( x^4 \) using \( x^2 \):
\[ x^4 = (x^2)^2 = (7 + 4\sqrt{3})^2 \]
\[ = 7^2 + 2 \cdot 7 \cdot 4\sqrt{3} + (4\sqrt{3})^2 \]
\[ = 49 + 56\sqrt{3} + 48 = 97 + 56\sqrt{3} \]

### Step 4: Substitute into the expression
Now we substitute \( x^4 \), \( x^3 \), \( x^2 \), and \( x \) into the expression \( x^4 - 4x^3 + x^2 + x + 1 \):
\[ x^4 - 4x^3 + x^2 + x + 1 = (97 + 56\sqrt{3}) - 4(26 + 15\sqrt{3}) + (7 + 4\sqrt{3}) + (2 + \sqrt{3}) + 1 \]

### Step 5: Simplify the expression
Calculating \( -4x^3 \):
\[ -4(26 + 15\sqrt{3}) = -104 - 60\sqrt{3} \]
Now substituting everything back:
\[ = (97 + 56\sqrt{3}) - (104 + 60\sqrt{3}) + (7 + 4\sqrt{3}) + (2 + \sqrt{3}) + 1 \]
Combining like terms:
\[ = (97 - 104 + 7 + 2 + 1) + (56\sqrt{3} - 60\sqrt{3} + 4\sqrt{3} + \sqrt{3}) \]
Calculating the constants:
\[ = (97 - 104 + 7 + 2 + 1) = 3 \]
Calculating the square root terms:
\[ = (56 - 60 + 4 + 1)\sqrt{3} = 1\sqrt{3} = \sqrt{3} \]
Thus, the final result is:
\[ = 3 + \sqrt{3} \]

### Final Answer
The value of \( x^4 - 4x^3 + x^2 + x + 1 \) is \( 3 + \sqrt{3} \).