Find the roots of the cubic equation x^3 + 4x^2 + 3x + 2 = 0.

The roots of the cubic equation x^3 + 4x^2 + 3x + 2 = 0 are to be found.

To find the roots of a cubic equation, we can use the Rational Root Theorem to identify possible rational roots. The Rational Root Theorem states that if a polynomial has integer coefficients, then any rational root of the polynomial must have a numerator that divides the constant term and a denominator that divides the leading coefficient.

In this case, the constant term is 2 and the leading coefficient is 1, so the possible rational roots are ±1, ±2. We can test these roots using synthetic division to see if they are actual roots of the equation.

Testing x = -1:
-1 | 1 4 3 2
| -1 -3 0
| 1 3 0
This tells us that x + 1 is a factor of the equation.

Testing x = -2:
-2 | 1 4 3 2
| -2 -4 2
| 1 2 -1
This tells us that x + 2 is also a factor of the equation.

Using these factors, we can write the equation as (x + 1)(x + 2)(x + k) = 0, where k is the remaining root. Expanding this gives us x^3 + 7x^2 + 14x + 8 = 0. We can then use synthetic division to find the value of k:
-2 | 1 7 14 8
| -2 -10 -8
| 1 5 4
This tells us that k = -4.

Therefore, the roots of the equation x^3 + 4x^2 + 3x + 2 = 0 are -1, -2, and -4.