To simplify the given expression, let's first rewrite it in a more readable format:

\[ A = \frac{\sqrt{2} + 1}{\sqrt{3} + 1} \div \frac{\sqrt{4} - 2\sqrt{3}}{\sqrt{2} - 1} \]

To divide by a fraction, we multiply by its reciprocal:

\[ A = \frac{\sqrt{2} + 1}{\sqrt{3} + 1} \times \frac{\sqrt{2} - 1}{\sqrt{4} - 2\sqrt{3}} \]

Next, let's simplify each part of the expression separately.

### Simplifying the first fraction:
\[ \frac{\sqrt{2} + 1}{\sqrt{3} + 1} \]

To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{3} - 1\):

\[ \frac{(\sqrt{2} + 1)(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} \]

The denominator simplifies using the difference of squares:

\[ (\sqrt{3} + 1)(\sqrt{3} - 1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2 \]

Now, expand the numerator:

\[ (\sqrt{2} + 1)(\sqrt{3} - 1) = \sqrt{2} \cdot \sqrt{3} - \sqrt{2} + \sqrt{3} - 1 = \sqrt{6} - \sqrt{2} + \sqrt{3} - 1 \]

So, the first fraction becomes:

\[ \frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} \]

### Simplifying the second fraction:
\[ \frac{\sqrt{2} - 1}{\sqrt{4} - 2\sqrt{3}} \]

First, simplify the denominator:

\[ \sqrt{4} - 2\sqrt{3} = 2 - 2\sqrt{3} \]

Factor out the common term in the denominator:

\[ 2(1 - \sqrt{3}) \]

So, the second fraction becomes:

\[ \frac{\sqrt{2} - 1}{2(1 - \sqrt{3})} \]

To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator, which is \(1 + \sqrt{3}\):

\[ \frac{(\sqrt{2} - 1)(1 + \sqrt{3})}{2(1 - \sqrt{3})(1 + \sqrt{3})} \]

The denominator simplifies using the difference of squares:

\[ 2((1)^2 - (\sqrt{3})^2) = 2(1 - 3) = 2(-2) = -4 \]

Now, expand the numerator:

\[ (\sqrt{2} - 1)(1 + \sqrt{3}) = \sqrt{2} \cdot 1 + \sqrt{2} \cdot \sqrt{3} - 1 \cdot 1 - 1 \cdot \sqrt{3} = \sqrt{2} + \sqrt{6} - 1 - \sqrt{3} \]

So, the second fraction becomes:

\[ \frac{\sqrt{2} + \sqrt{6} - 1 - \sqrt{3}}{-4} \]

### Combining the results:

Now, we have:

\[ A = \left( \frac{\sqrt{6} - \sqrt{2} + \sqrt{3} - 1}{2} \right) \times \left( \frac{\sqrt{2} + \sqrt{6} - 1 - \sqrt{3}}{-4} \right) \]

Combine the fractions:

\[ A = \frac{(\sqrt{6} - \sqrt{2} + \sqrt{3} - 1)(\sqrt{2} + \sqrt{6} - 1 - \sqrt{3})}{2 \times -4} \]

\[ A = \frac{(\sqrt{6} - \sqrt{2} + \sqrt{3} - 1)(\sqrt{2} + \sqrt{6} - 1 - \sqrt{3})}{-8} \]

This expression is quite complex, and further simplification might not lead to a more insightful form. However, this is the simplified form of the given expression.