Let's simplify the expressions for \( A \) and \( a \) step by step.

### For \( A \):
1. **First expression**:
\[ A = \sqrt{6} - 2\sqrt{5} + \sqrt{9} - 4\sqrt{5} \]
Simplifying \( \sqrt{9} \):
\[ \sqrt{9} = 3 \]
So,
\[ A = \sqrt{6} - 2\sqrt{5} + 3 - 4\sqrt{5} = \sqrt{6} + 3 - 6\sqrt{5} \]

2. **Second expression**:
\[ A = \sqrt{7} - 4\sqrt{3} - \sqrt{4} - 2\sqrt{3} \]
Simplifying \( \sqrt{4} \):
\[ \sqrt{4} = 2 \]
So,
\[ A = \sqrt{7} - 4\sqrt{3} - 2 - 2\sqrt{3} = \sqrt{7} - 2 - 6\sqrt{3} \]

From the above, we simplified both expressions for \( A \):
- \( A = \sqrt{6} + 3 - 6\sqrt{5} \)
- \( A = \sqrt{7} - 2 - 6\sqrt{3} \)

### Now for \( a \):
1. **First expression**:
\[ a = \sqrt{4} - 2\sqrt{3} + \sqrt{4} + 2\sqrt{3} \]
Simplifying \( \sqrt{4} \):
\[ \sqrt{4} = 2 \]
So,
\[ a = 2 - 2\sqrt{3} + 2 + 2\sqrt{3} = 4 \]

2. **Second expression**:
\[ a = \sqrt{4} + 2\sqrt{3} - \sqrt{4} - 2\sqrt{3} \]
Again simplifying \( \sqrt{4} \):
\[ a = 2 + 2\sqrt{3} - 2 - 2\sqrt{3} = 0 \]

3. **Third expression**:
\[ a = \sqrt{3} + 2\sqrt{2} + \sqrt{3} - 2\sqrt{2} \]
So,
\[ a = 2\sqrt{3} + 2\sqrt{2} - 2\sqrt{2} = 2\sqrt{3} \]

4. **Fourth expression**:
\[ a = \sqrt{8} - 2\sqrt{7} - \sqrt{8} + 2\sqrt{7} \]
Simplifying \( \sqrt{8} \):
\[ a = 2\sqrt{2} - 2\sqrt{7} - 2\sqrt{2} + 2\sqrt{7} = 0 \]

5. **Fifth expression**:
\[ a = \sqrt{28} + 10\sqrt{3} + \sqrt{19} - 8\sqrt{3} \]
Simplifying \( \sqrt{28} \):
\[ \sqrt{28} = 2\sqrt{7} \]
So,
\[ a = 2\sqrt{7} + 10\sqrt{3} - 8\sqrt{3} + \sqrt{19} = 2\sqrt{7} + 2\sqrt{3} + \sqrt{19} \]

### Summary of simplified expressions:
- For \( A \):
- \( A = \sqrt{6} + 3 - 6\sqrt{5} \)
- \( A = \sqrt{7} - 2 - 6\sqrt{3} \)

- For \( a \):
- \( a = 4 \)
- \( a = 0 \)
- \( a = 2\sqrt{3} \)
- \( a = 0 \)
- \( a = 2\sqrt{7} + 2\sqrt{3} + \sqrt{19} \)