This equation is a cubic polynomial, meaning it is of the form ax^3 + bx^2 + cx + d = 0.

In this case, the coefficients are:
a = 3
b = -2
c = 1
d = -10

To solve this equation, we can use various methods such as factoring, synthetic division, or the cubic formula. Let's try factoring first:

3x^3 - 2x^2 + x - 10 = 0
Rearrange the equation:
3x^3 - 6x^2 + 4x - 8x^2 + 8x - 10 = 0
Group terms:
3x^2(x - 2) + 4x(x - 2) - 10 = 0
Factor out common terms:
(3x^2 + 4x)(x - 2) - 10 = 0
Factor further:
x(3x + 4)(x - 2) - 10 = 0

Now, set each factor to zero and solve for x:
1) x = 0
2) 3x + 4 = 0
3x = -4
x = -4/3
3) x - 2 = 0
x = 2

Therefore, the solutions to the equation 3x^3 - 2x^2 + x - 10 = 0 are x = 0, x = -4/3, and x = 2.