To rationalize the denominator of the expression \(-\frac{6}{\sqrt{5} + 3 + \sqrt{2}}\), we will follow a systematic approach.
Step 1: Identify the Conjugate
- The denominator is \(\sqrt{5} + 3 + \sqrt{2}\).
- To rationalize, we need to multiply by the conjugate, which is \(\sqrt{5} + 3 - \sqrt{2}\).
Step 2: Multiply Numerator and Denominator
- Multiply both the numerator and the denominator by the conjugate:
\[
-\frac{6}{\sqrt{5} + 3 + \sqrt{2}} \cdot \frac{\sqrt{5} + 3 - \sqrt{2}}{\sqrt{5} + 3 - \sqrt{2}}
\]
- This results in:
\[
\frac{-6(\sqrt{5} + 3 - \sqrt{2})}{(\sqrt{5} + 3 + \sqrt{2})(\sqrt{5} + 3 - \sqrt{2})}
\]
Step 3: Simplify the Denominator
- The denominator simplifies using the difference of squares formula:
\[
(\sqrt{5} + 3)^2 - (\sqrt{2})^2
\]
- Calculating it gives:
\[
(5 + 6 + 9) - 2 = 18
\]
Step 4: Expand the Numerator
- Expand the numerator:
\[
-6(\sqrt{5} + 3 - \sqrt{2}) = -6\sqrt{5} - 18 + 6\sqrt{2}
\]
Final Expression
- Putting it all together, we have:
\[
\frac{-6\sqrt{5} - 18 + 6\sqrt{2}}{18}
\]
- This can be simplified further by dividing each term by 18:
\[
-\frac{\sqrt{5}}{3} - 1 + \frac{\sqrt{2}}{3}
\]
Thus, the rationalized form of the original expression is:
\[
-\frac{\sqrt{5}}{3} + \frac{\sqrt{2}}{3} - 1
\]