Rút gọn biểu thức có mẫu thức chứa hằng đẳng thức

To simplify the expressions in the image, we can use algebraic techniques, focusing on rationalizing the denominators and applying algebraic identities where necessary. Here’s a general approach to tackle the provided problems:

### Part a:

1. **Expression A:**
\[
A = \frac{x - 4}{\sqrt{x - 2}} \quad \text{(rationalize the denominator)}
\]

2. **Expression B:**
\[
B = \frac{x + 3}{\sqrt{x + 3} - \sqrt{x - 3}} \quad \text{(multiply by the conjugate)}
\]

3. **Expression M:**
\[
M = \frac{2\sqrt{x} + 1}{\sqrt{x + 3} - 3} \quad \text{(rationalize the denominator)}
\]

### Part b:

1. **Expression A:**
\[
A = \frac{x + 2 + \sqrt{x + 1}}{\sqrt{x - 1}} \quad \text{(multiply numerator and denominator by conjugate)}
\]

2. **Expression P:**
\[
P = \frac{x + 1}{x\sqrt{x}} \quad \text{(simplify before rationalizing)}
\]

3. **Expression M:**
\[
M = \frac{x^2 - \sqrt{x}}{x + 1} \quad \text{(factor if possible)}
\]

### Part c:

1. **Expression B:**
\[
B = \left( \frac{1 + \sqrt{x}}{2\sqrt{x} - 3} \right) \quad \text{(simplify)}
\]

2. **Expression M:**
\[
M = \left( \frac{x - 2}{\sqrt{x + 3\sqrt{2}}} \right) \quad \text{(apply similar techniques)}
\]

3. **Expression M:**
\[
M = \sqrt{x} - \frac{1}{3-\sqrt{6}} \quad \text{(rationalize the denominator)}
\]

### General Steps:
- **Factor and Cancel:** Factor out common terms in numerators and denominators.
- **Rationalize:** Multiply by the conjugate when dealing with radical expressions.
- **Combine Terms:** When dealing with fractions, find a common denominator to combine.

If you have specific questions about certain expressions or if you want detailed steps for a specific one, please let me know!