Understanding the Equation
The given equation is a quartic equation, which means it is a polynomial equation of degree 4. The equation is as follows:

x^4 + x^3 - 3x^2 - x + 2 = 0

To find the number of distinct real roots of this equation, we need to analyze its behavior and properties.

Using Descartes' Rule of Signs
Descartes' Rule of Signs is a useful tool to determine the number of positive and negative real roots of a polynomial equation by analyzing the signs of its coefficients.

In our equation, if we look at the coefficients, we can see that the signs alternate:

+1, +1, -3, -1, +2

According to Descartes' Rule of Signs, the number of positive real roots is either equal to the number of sign changes or less than it by an even number. In this case, there is one sign change, so there is either one positive real root or three positive real roots.

Similarly, the number of negative real roots is either equal to the number of sign changes or less than it by an even number. In this case, there are two sign changes, so there are either two negative real roots or no negative real roots.

Using the Intermediate Value Theorem
The Intermediate Value Theorem states that if a polynomial function f(x) is continuous on the interval [a, b], and f(a) and f(b) have opposite signs, then there exists at least one real root of f(x) between a and b.

To apply this theorem, we need to find intervals where the function changes sign. By plotting the function or using a graphing software, we can observe the following sign changes:

Between -2 and -1: from positive to negative
Between -1 and 0: from negative to positive
Between 1 and 2: from negative to positive

Since there are three sign changes, there must be three distinct real roots within these intervals. Therefore, the value of n, which represents the number of distinct real roots, is 3.

Conclusion
The given quartic equation has three distinct real roots. We determined this by applying Descartes' Rule of Signs to analyze the signs of the coefficients and using the Intermediate Value Theorem to find intervals where the function changes sign. By following these methods, we can confidently conclude that the number of distinct real roots is 3.