To rationalize the denominator of the expression $\frac{-1}{5 + \sqrt{11}}$, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $5 + \sqrt{11}$ is $5 - \sqrt{11}$.

Steps:

1. Expression: $\frac{-1}{5 + \sqrt{11}}$

2. Multiply numerator and denominator by the conjugate $5 - \sqrt{11}$: $\frac{-1}{5 + \sqrt{11}} \times \frac{5 - \sqrt{11}}{5 - \sqrt{11}}$

3. Simplify the denominator using the difference of squares: $(5 + \sqrt{11})(5 - \sqrt{11}) = 5^2 - (\sqrt{11})^2 = 25 - 11 = 14$

4. Simplify the numerator: $(-1) \times (5 - \sqrt{11}) = -(5 - \sqrt{11}) = -5 + \sqrt{11}$

5. Final result: $\frac{-5 + \sqrt{11}}{14}$

So, the rationalized expression is:

$\frac{-5 + \sqrt{11}}{14}$

Would you like further details or clarifications on this process? Here are some related questions:

1. What is the conjugate of a binomial, and why is it used in rationalization?
2. How do you apply the difference of squares formula to simplify expressions?
3. What would happen if you didn't rationalize the denominator?
4. How do you handle expressions with square roots in both the numerator and denominator?
5. Can this method of rationalization be extended to more complex expressions, like cube roots?

Tip: When rationalizing, always remember that multiplying by the conjugate eliminates the radical in the denominator.