To simplify the expression \( \frac{4}{-3 - \sqrt{2}} \) and rationalize the denominator, we need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \( -3 - \sqrt{2} \) is \( -3 + \sqrt{2} \).
Here's how to proceed:
- Multiply numerator and denominator by the conjugate:
\[ \frac{4}{-3 - \sqrt{2}} \cdot \frac{-3 + \sqrt{2}}{-3 + \sqrt{2}} = \frac{4(-3 + \sqrt{2})}{(-3 - \sqrt{2})(-3 + \sqrt{2})} \]
- Calculate the denominator using the difference of squares:
\[ (-3 - \sqrt{2})(-3 + \sqrt{2}) = (-3)^2 - (\sqrt{2})^2 = 9 - 2 = 7 \]
- Calculate the numerator:
\[ 4(-3 + \sqrt{2}) = -12 + 4\sqrt{2} \]
- Combine results:
Putting the simplified numerator and denominator together gives us:
\[ \frac{-12 + 4\sqrt{2}}{7} \]
Thus, the simplified expression is:
\[ \frac{4\sqrt{2} - 12}{7} \]
This is the final answer.