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

Explanation

To find the value of \( x^4 - 4x^3 + x^2 + x + 1 \) given \( x = 2 + \sqrt{3} \), we will first calculate \( x^2 \), \( x^3 \), and \( x^4 \) step by step.

### Step 1: Calculate \( x^2 \)

\[ x^2 = (2 + \sqrt{3})^2 \]
Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \):
\[ x^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 \]
\[ = 4 + 4\sqrt{3} + 3 \]
\[ = 7 + 4\sqrt{3} \]

### Step 2: Calculate \( x^3 \)

\[ x^3 = x \cdot x^2 = (2 + \sqrt{3})(7 + 4\sqrt{3}) \]
Using the distributive property:
\[ = 2 \cdot 7 + 2 \cdot 4\sqrt{3} + \sqrt{3} \cdot 7 + \sqrt{3} \cdot 4\sqrt{3} \]
\[ = 14 + 8\sqrt{3} + 7\sqrt{3} + 4 \cdot 3 \]
\[ = 14 + 8\sqrt{3} + 7\sqrt{3} + 12 \]
\[ = 26 + 15\sqrt{3} \]

### Step 3: Calculate \( x^4 \)

\[ x^4 = x \cdot x^3 = (2 + \sqrt{3})(26 + 15\sqrt{3}) \]
Using the distributive property:
\[ = 2 \cdot 26 + 2 \cdot 15\sqrt{3} + \sqrt{3} \cdot 26 + \sqrt{3} \cdot 15\sqrt{3} \]
\[ = 52 + 30\sqrt{3} + 26\sqrt{3} + 15 \cdot 3 \]
\[ = 52 + 30\sqrt{3} + 26\sqrt{3} + 45 \]
\[ = 97 + 56\sqrt{3} \]

### Step 4: Substitute into the expression

Now we substitute \( x^4 \), \( x^3 \), \( x^2 \), and \( x \) into the expression \( x^4 - 4x^3 + x^2 + x + 1 \):

\[ x^4 - 4x^3 + x^2 + x + 1 = (97 + 56\sqrt{3}) - 4(26 + 15\sqrt{3}) + (7 + 4\sqrt{3}) + (2 + \sqrt{3}) + 1 \]

Calculating \( -4x^3 \):
\[ -4(26 + 15\sqrt{3}) = -104 - 60\sqrt{3} \]

Now substituting back:
\[ = (97 + 56\sqrt{3}) - (104 + 60\sqrt{3}) + (7 + 4\sqrt{3}) + (2 + \sqrt{3}) + 1 \]

### Step 5: Combine like terms

Combining the constant terms:
\[ 97 - 104 + 7 + 2 + 1 = 3 \]

Combining the terms with \( \sqrt{3} \):
\[ 56\sqrt{3} - 60\sqrt{3} + 4\sqrt{3} + \sqrt{3} = (56 - 60 + 4 + 1)\sqrt{3} = 1\sqrt{3} \]

### Final Result

Thus, the expression simplifies to:
\[ 3 + \sqrt{3} \]