To calculate \( A = (4 + \sqrt{15})(\sqrt{10} - \sqrt{6})\sqrt{(4 - \sqrt{15})} \), we can break it down into steps.

### Step 1: Simplify \( \sqrt{10} - \sqrt{6} \)

This expression is already in its simplest form, so we leave it as is.

### Step 2: Calculate \( 4 - \sqrt{15} \)

This expression also cannot be simplified further.

### Step 3: Calculate \( \sqrt{(4 - \sqrt{15})} \)

Let’s leave this term as \( \sqrt{(4 - \sqrt{15})} \).

### Step 4: Combine everything

Now we can expand \( (4 + \sqrt{15})(\sqrt{10} - \sqrt{6}) \):

\[
A = (4\sqrt{10} - 4\sqrt{6} + \sqrt{15}\sqrt{10} - \sqrt{15}\sqrt{6}) \cdot \sqrt{(4 - \sqrt{15})}
\]

Now, let's simplify further:

1. **Calculate** \( 4\sqrt{10} - 4\sqrt{6} = 4(\sqrt{10} - \sqrt{6}) \).
2. **Calculate** \( \sqrt{15}\sqrt{10} - \sqrt{15}\sqrt{6} = \sqrt{15}(\sqrt{10} - \sqrt{6}) \).

Combining these:

\[
A = \left(4 + \sqrt{15}\right)(\sqrt{10} - \sqrt{6}) \cdot \sqrt{(4 - \sqrt{15})}
\]

### Final Expression

The final form of \( A \) is:

\[
A = \left(4 + \sqrt{15}\right)(\sqrt{10} - \sqrt{6}) \cdot \sqrt{(4 - \sqrt{15})}
\]

If you need to calculate a numerical approximation, you can plug in values for \( \sqrt{10} \), \( \sqrt{6} \), and \( \sqrt{15} \) to arrive at a numerical answer.

Feel free to ask if you have further questions!