If x `=(sqrt(5)-2)/(sqrt(5)+2)` and `y = (sqrt(5)+2)/(sqrt(5)-2)` : find :
`x^(2)+ y^(2)+xy `

Answer

To solve the problem, we need to find the value of \( x^2 + y^2 + xy \), where \( x = \frac{\sqrt{5}-2}{\sqrt{5}+2} \) and \( y = \frac{\sqrt{5}+2}{\sqrt{5}-2} \).

### Step 1: Simplify \( x \)

We start with the expression for \( x \):
\[ x = \frac{\sqrt{5}-2}{\sqrt{5}+2} \]

To simplify \( x \), we multiply the numerator and denominator by the conjugate of the denominator, which is \( \sqrt{5}-2 \):
\[ x = \frac{(\sqrt{5}-2)(\sqrt{5}-2)}{(\sqrt{5}+2)(\sqrt{5}-2)} \]

### Step 2: Calculate the Denominator

The denominator can be calculated using the difference of squares:
\[ (\sqrt{5}+2)(\sqrt{5}-2) = (\sqrt{5})^2 - (2)^2 = 5 - 4 = 1 \]

### Step 3: Calculate the Numerator

Now, calculate the numerator:
\[ (\sqrt{5}-2)(\sqrt{5}-2) = (\sqrt{5})^2 - 2 \cdot \sqrt{5} \cdot 2 + (2)^2 = 5 - 4\sqrt{5} + 4 = 9 - 4\sqrt{5} \]

### Step 4: Final Simplification of \( x \)

Thus, we have:
\[ x = \frac{9 - 4\sqrt{5}}{1} = 9 - 4\sqrt{5} \]

### Step 5: Simplify \( y \)

Now, we simplify \( y \):
\[ y = \frac{\sqrt{5}+2}{\sqrt{5}-2} \]

Again, we multiply by the conjugate of the denominator:
\[ y = \frac{(\sqrt{5}+2)(\sqrt{5}+2)}{(\sqrt{5}-2)(\sqrt{5}+2)} \]

### Step 6: Calculate the Denominator for \( y \)

The denominator is the same as before:
\[ (\sqrt{5}-2)(\sqrt{5}+2) = 1 \]

### Step 7: Calculate the Numerator for \( y \)

Now, calculate the numerator:
\[ (\sqrt{5}+2)(\sqrt{5}+2) = (\sqrt{5})^2 + 2 \cdot \sqrt{5} \cdot 2 + (2)^2 = 5 + 4\sqrt{5} + 4 = 9 + 4\sqrt{5} \]

### Step 8: Final Simplification of \( y \)

Thus, we have:
\[ y = \frac{9 + 4\sqrt{5}}{1} = 9 + 4\sqrt{5} \]

### Step 9: Calculate \( x^2 \)

Now we calculate \( x^2 \):
\[ x^2 = (9 - 4\sqrt{5})^2 = 9^2 - 2 \cdot 9 \cdot 4\sqrt{5} + (4\sqrt{5})^2 \]
\[ = 81 - 72\sqrt{5} + 16 \cdot 5 = 81 - 72\sqrt{5} + 80 = 161 - 72\sqrt{5} \]

### Step 10: Calculate \( y^2 \)

Now we calculate \( y^2 \):
\[ y^2 = (9 + 4\sqrt{5})^2 = 9^2 + 2 \cdot 9 \cdot 4\sqrt{5} + (4\sqrt{5})^2 \]
\[ = 81 + 72\sqrt{5} + 80 = 161 + 72\sqrt{5} \]

### Step 11: Calculate \( xy \)

Now we calculate \( xy \):
\[ xy = (9 - 4\sqrt{5})(9 + 4\sqrt{5}) = 9^2 - (4\sqrt{5})^2 = 81 - 80 = 1 \]

### Step 12: Combine Results

Finally, we combine the results to find \( x^2 + y^2 + xy \):
\[ x^2 + y^2 + xy = (161 - 72\sqrt{5}) + (161 + 72\sqrt{5}) + 1 \]
\[ = 161 + 161 + 1 = 323 \]

Thus, the final answer is:
\[ \boxed{323} \]