The equation given is:

$x^3 - 5x^2 - x + 21 = 0$

To solve this graphically using a graphing calculator, follow these steps:

1. Enter the function $f(x) = x^3 - 5x^2 - x + 21$ into the graphing calculator.
2. Plot the graph of this cubic equation.
3. Identify the points where the graph crosses the x-axis. These points correspond to the real solutions of the equation.
4. The x-values at these intersections are the real solutions to the equation.

Would you like me to compute the real solutions for this equation?

Here are some follow-up questions you can explore:

1. How many solutions does a cubic equation typically have?
2. What is the significance of the points where the graph crosses the x-axis?
3. Can there be both real and complex solutions for a cubic equation?
4. How does the shape of a cubic function differ from quadratic or linear functions?
5. How would this process change if the equation had no real solutions?

Tip: For polynomials, the degree of the equation often indicates the maximum number of possible real solutions.