Let's consider the expressions given in the problem:
[tex]\[ x = \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} \][/tex]
[tex]\[ y = \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} \][/tex]
We need to find the value of [tex]\( x^2 + x + y + y^2 \)[/tex].
First, let's find [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex].
### Step 1: Calculating [tex]\( x^2 \)[/tex]
[tex]\[ x = \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} \][/tex]
[tex]\[ x^2 = \left(\frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}}\right)^2 \][/tex]
[tex]\[ x^2 = \frac{(\sqrt{5} - \sqrt{2})^2}{(\sqrt{5} + \sqrt{2})^2} \][/tex]
### Step 2: Calculating [tex]\( y^2 \)[/tex]
[tex]\[ y = \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} \][/tex]
[tex]\[ y^2 = \left(\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}\right)^2 \][/tex]
[tex]\[ y^2 = \frac{(\sqrt{5} + \sqrt{2})^2}{(\sqrt{5} - \sqrt{2})^2} \][/tex]
### Step 3: Calculating [tex]\( x + y \)[/tex]
[tex]\[ x = \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} \][/tex]
[tex]\[ y = \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} \][/tex]
[tex]\[ x + y = \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} + \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} \][/tex]
### Step 4: Summing Up [tex]\( x^2 + x + y + y^2 \)[/tex]
Now let's add them together to find the required expression.
[tex]\[ x^2 + x + y + y^2 = \frac{(\sqrt{5} - \sqrt{2})^2}{(\sqrt{5} + \sqrt{2})^2} + \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} + \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} + \frac{(\sqrt{5} + \sqrt{2})^2}{(\sqrt{5} - \sqrt{2})^2} \][/tex]
So, we have:
[tex]\[ x^2 = \frac{(\sqrt{2} - \sqrt{5})^2}{(\sqrt{2} + \sqrt{5})^2} \][/tex]
[tex]\[ y^2 = \frac{(\sqrt{2} + \sqrt{5})^2}{(\sqrt{2} - \sqrt{5})^2} \][/tex]
And:
[tex]\[ x + y = \frac{(\sqrt{2} - \sqrt{5})}{(\sqrt{2} + \sqrt{5})} + \frac{(\sqrt{2} + \sqrt{5})}{(\sqrt{2} - \sqrt{5})} \][/tex]
Thus, the desired sum is:
[tex]\[ x^2 + x + y + y^2 = \frac{(\sqrt{2} - \sqrt{5})^2}{(\sqrt{2} + \sqrt{5})^2} + \frac{\sqrt{2} - \sqrt{5}}{\(\sqrt{2} + \sqrt{5}\)} + \frac{\sqrt{2} + \sqrt{5}}{\(\sqrt{2} - \sqrt{5}\)} + \frac{(\sqrt{2} + \sqrt{5})^2}{(\sqrt{2} - \sqrt{5})^2} \][/tex]
Which evaluates to:
[tex]\[ (-\sqrt{2} + \sqrt{5})^2/(\sqrt{2} + \sqrt{5})^2 + (-\sqrt{2} + \sqrt{5})/(\sqrt{2} + \sqrt{5}) + (\sqrt{2} + \sqrt{5})/(-\sqrt{2} + \sqrt{5}) + (\sqrt{2} + \sqrt{5})^2/(-\sqrt{2} + \sqrt{5})^2 \][/tex]
This is the required result.
