To simplify the given expression \( A \), we can rationalize the denominators of each fraction. The given expression is:

\[ A = \frac{1}{2 - \sqrt{2}} - \frac{1}{2 + \sqrt{2}} + \frac{\sqrt{15} - 3\sqrt{3}}{3 - \sqrt{5}} \]

### Step 1: Rationalizing the first fraction \(\frac{1}{2 - \sqrt{2}}\)

Multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{1}{2 - \sqrt{2}} \cdot \frac{2 + \sqrt{2}}{2 + \sqrt{2}} = \frac{2 + \sqrt{2}}{4 - 2} = \frac{2 + \sqrt{2}}{2} = 1 + \frac{\sqrt{2}}{2} \]

### Step 2: Rationalizing the second fraction \(\frac{1}{2 + \sqrt{2}}\)

Similarly, multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{1}{2 + \sqrt{2}} \cdot \frac{2 - \sqrt{2}}{2 - \sqrt{2}} = \frac{2 - \sqrt{2}}{4 - 2} = \frac{2 - \sqrt{2}}{2} = 1 - \frac{\sqrt{2}}{2} \]

### Step 3: Simplifying the first two terms

Now, subtract the two simplified expressions:

\[ \left(1 + \frac{\sqrt{2}}{2}\right) - \left(1 - \frac{\sqrt{2}}{2}\right) = 1 + \frac{\sqrt{2}}{2} - 1 + \frac{\sqrt{2}}{2} = \sqrt{2} \]

### Step 4: Rationalizing the third fraction \(\frac{\sqrt{15} - 3\sqrt{3}}{3 - \sqrt{5}}\)

Multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{\sqrt{15} - 3\sqrt{3}}{3 - \sqrt{5}} \cdot \frac{3 + \sqrt{5}}{3 + \sqrt{5}} = \frac{(\sqrt{15} - 3\sqrt{3})(3 + \sqrt{5})}{3^2 - (\sqrt{5})^2} = \frac{(\sqrt{15})(3) + (\sqrt{15})(\sqrt{5}) - (3\sqrt{3})(3) - (3\sqrt{3})(\sqrt{5})}{9 - 5} \]

\[ = \frac{3\sqrt{15} + \sqrt{75} - 9\sqrt{3} - 3\sqrt{15}}{4} = \frac{\sqrt{75} - 9\sqrt{3}}{4} \]

Knowing that \(\sqrt{75} = 5\sqrt{3}\):

\[ \frac{5\sqrt{3} - 9\sqrt{3}}{4} = \frac{-4\sqrt{3}}{4} = -\sqrt{3} \]

### Step 5: Combining all simplified terms

Now, add the simplified terms:

\[ \sqrt{2} - \sqrt{3} \]

Thus, the simplified form of \( A \) is:

\[ A = \sqrt{2} - \sqrt{3} \]