If `p = (sqrt(3) - 2)/(sqrt(2) + 1)` , then which one of the following equals `p - 4`?

Options

A. `sqrt(3) - 2`
B. `sqrt(3) + 2`
C. 2
D. `-2sqrt(2) + sqrt(6) - sqrt(3) - 2`

Answer

To solve the problem, we need to find the expression for \( p - 4 \) where \( p = \frac{\sqrt{3} - 2}{\sqrt{2} + 1} \).

### Step-by-Step Solution:

1. **Start with the expression for \( p \)**:
\[ p = \frac{\sqrt{3} - 2}{\sqrt{2} + 1} \]

2. **Write the expression for \( p - 4 \)**:
\[ p - 4 = \frac{\sqrt{3} - 2}{\sqrt{2} + 1} - 4 \]

3. **Combine the terms over a common denominator**:
To combine the terms, we will rewrite \( 4 \) with the denominator \( \sqrt{2} + 1 \):
\[ p - 4 = \frac{\sqrt{3} - 2}{\sqrt{2} + 1} - \frac{4(\sqrt{2} + 1)}{\sqrt{2} + 1} \]
This simplifies to:
\[ p - 4 = \frac{\sqrt{3} - 2 - 4\sqrt{2} - 4}{\sqrt{2} + 1} \]
\[ = \frac{\sqrt{3} - 4\sqrt{2} - 6}{\sqrt{2} + 1} \]

4. **Multiply by the conjugate of the denominator**:
The conjugate of \( \sqrt{2} + 1 \) is \( \sqrt{2} - 1 \). We multiply both the numerator and denominator by this conjugate:
\[ p - 4 = \frac{(\sqrt{3} - 4\sqrt{2} - 6)(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)} \]

5. **Simplify the denominator**:
The denominator simplifies as follows:
\[ (\sqrt{2} + 1)(\sqrt{2} - 1) = 2 - 1 = 1 \]

6. **Expand the numerator**:
Now we expand the numerator:
\[ = (\sqrt{3} - 4\sqrt{2} - 6)(\sqrt{2} - 1) \]
\[ = \sqrt{3}\sqrt{2} - \sqrt{3} - 4\sqrt{2}\sqrt{2} + 4\sqrt{2} - 6\sqrt{2} + 6 \]
\[ = \sqrt{6} - \sqrt{3} - 8 + 4\sqrt{2} - 6\sqrt{2} + 6 \]
\[ = \sqrt{6} - \sqrt{3} - 2\sqrt{2} - 2 \]

7. **Final expression**:
Thus, we have:
\[ p - 4 = \sqrt{6} - \sqrt{3} - 2\sqrt{2} - 2 \]

### Final Result:
The expression for \( p - 4 \) is:
\[ p - 4 = \sqrt{6} - \sqrt{3} - 2\sqrt{2} - 2 \]