The approximate solutions for the equation $-x^3 + 5x + 6 = 0$ are:

$x_1 \approx -1.29, \quad x_2 \approx 2.69$

These values are the estimates for the roots where $f(x) = -4$. The function crosses the x-axis near these points.

Would you like more details or further refinement?

Here are 5 related questions you might explore:

1. How do we exactly find the roots of a cubic equation?
2. What is the behavior of cubic functions as $x$ approaches infinity?
3. How can we verify if a value is an exact solution for a cubic equation?
4. What methods exist to find the roots of higher-degree polynomials?
5. How does graphing help estimate the solutions of equations?

Tip: Checking solutions by substitution can help verify the accuracy of the approximations.