To simplify the expression

\[
P = \left( \frac{\sqrt{x} + y\sqrt{y}}{\sqrt{x} + \sqrt{y}} - \frac{y\sqrt{y} - y\sqrt{x}}{\sqrt{y} - \sqrt{x}} \right) \cdot \frac{\sqrt{x} - \sqrt{y}}{\sqrt{x} + \sqrt{y}}
\]

we can follow these steps:

1. Simplify each term in the expression separately.

### Step 1: Simplifying the first fraction

The first term is

\[
\frac{\sqrt{x} + y\sqrt{y}}{\sqrt{x} + \sqrt{y}}
\]

This cannot be simplified directly, but we will revisit it after looking at the second fraction.

### Step 2: Simplifying the second fraction

The second term is:

\[
-\frac{y\sqrt{y} - y\sqrt{x}}{\sqrt{y} - \sqrt{x}}
\]

Notice that it can be rewritten as:

\[
-\frac{y(\sqrt{y} - \sqrt{x})}{\sqrt{y} - \sqrt{x}} = -y
\]

### Step 3: Combine terms

Now we substitute this back into the expression for \(P\):

\[
P = \left( \frac{\sqrt{x} + y\sqrt{y}}{\sqrt{x} + \sqrt{y}} + y \right) \cdot \frac{\sqrt{x} - \sqrt{y}}{\sqrt{x} + \sqrt{y}}
\]

### Step 4: Combine terms in the parentheses

Now we need to combine the two terms in the parentheses:

\[
\frac{\sqrt{x} + y\sqrt{y}}{\sqrt{x} + \sqrt{y}} + y = \frac{\sqrt{x} + y\sqrt{y} + y(\sqrt{x} + \sqrt{y})}{\sqrt{x} + \sqrt{y}}
\]

This simplifies to:

\[
\frac{\sqrt{x} + y\sqrt{y} + y\sqrt{x} + y\sqrt{y}}{\sqrt{x} + \sqrt{y}} = \frac{(1 + y)\sqrt{x} + (1 + y)\sqrt{y}}{\sqrt{x} + \sqrt{y}}
\]

### Step 5: Final Expression

Combining this result we have:

\[
P = \frac{(1 + y)(\sqrt{x} + \sqrt{y})}{\sqrt{x} + \sqrt{y}} \cdot \frac{\sqrt{x} - \sqrt{y}}{\sqrt{x} + \sqrt{y}}
\]

The \((\sqrt{x} + \sqrt{y})\) cancels out (assuming \(\sqrt{x} + \sqrt{y} \neq 0\)):

\[
P = (1 + y)(\sqrt{x} - \sqrt{y})
\]

Thus, the simplified expression is:

\[
P = (1 + y)(\sqrt{x} - \sqrt{y})
\]