We can use the Fundamental Theorem of Algebra to determine the number of complex roots of the polynomial F(x) = x^4 + x^3 - 5x^2 + x - 6. According to this theorem, a polynomial of degree n has exactly n complex roots (counting multiplicities).
To find the number of real roots, we can use Descartes' Rule of Signs. We count the number of sign changes in the coefficients of F(x) and F(-x), which gives us the upper bound on the number of positive and negative real roots, respectively.
F(x) = x^4 + x^3 - 5x^2 + x - 6 has two sign changes, so it has either 2 or 0 positive real roots.
F(-x) = -x^4 + x^3 - 5x^2 - x - 6 has one sign change, so it has either 1 or 3 negative real roots.
Therefore, F(x) has either 0, 1, or 2 real roots.
Since F(x) is a polynomial of degree 4, it has exactly 4 complex roots (counting multiplicities). Therefore, F(x) has either 2 complex conjugate pairs of roots or 4 distinct roots.
In summary, F(x) has either 0, 1, or 2 real roots, and either 2 complex conjugate pairs of roots or 4 distinct roots.